NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 1 of 31

Stage 5: Mathematics STEM Build Make Create Unit 2 sample program

Overview Duration

In Build Make Create students will recall, re-learn and develop the following essential skills:

describe objects correctly: size, shape and position

describe liquids correctly: volume, mixtures

use and interpret units of measurement

understand the dimensions: length, area, volume, capacity

interpret and apply scale

In Build Make Create students will develop the following essential STEM understandings:

between the initial idea and final result, mathematics often provides the answer or information needed for each new step through a project

mathematics allows for safe testing of an idea by calculating expected outcomes based on scientific knowledge, creating scale models, creating digital simulations and mathematical modelling

mathematical terminology and diagrams allow people to communicate clearly and specifically about ideas and provide a common language for evaluating those ideas.

In Build Make Create students may have the opportunity to consider:

Which industries build, make and create?

Which Vocational Education and Training courses provide pathways to build, make and create?

What mathematics is needed in those courses and later on in related industries?

15 weeks.

Outcomes

A student:

applies Pythagoras’ theorem to calculate side lengths in right-angled triangles, and solves related problems (MA4-16MG)

calculates the areas of composite shapes, and the surface areas of rectangular and triangular prisms (MA5.1-8MG)

applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression (MA5.1-10MG)

describes and applies the properties of similar figures and scale drawings (MA5.1-11MG)

operates with ratios and rates, and explores their graphical representation (MA4-7NA)

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 2 of 31

uses appropriate terminology, diagrams and symbols in mathematical contexts (MA5.1-1WM)

selects and uses appropriate strategies to solve problems (MA5.1-2WM)

Language/Literacy STEM/VET

Enlargement and reduction.

Scale, scale factor and proportion

Mathematically “Similar” compared to common use of the word.

Edge, vertex, face.

Dimensions: 1D, 2D & 3D.

Interchangeability of height, breadth, width and depth in area and volume formulae.

Science: Retail, Hospitality. Primary Industries

Technology: Electrotechnology, Entertainment Industry, Information and Digital Technology, Primary Industries

Engineering: Metal and Engineering, Automotive, Construction

Strategies to support learning for Aboriginal and/or Torres Strait Islander students

Involve local Aboriginal communities and/or appropriate knowledge holders in determining suitable resources, or use Aboriginal or Torres Strait Islander authored or endorsed publications.

The “8 Ways of Learning for Aboriginal students” are one set of described strategies and these are incorporated throughout this Teaching and Learning Program. As described at 8ways.wikispaces.com, these can be summarised as: Tell a story. Make a plan. Think and do. Draw it. Take it outside. Try a new way. Watch first, then do. Share it with others.

Content Teaching & Learning STEM Resources & Stimulus

use the enlargement transformation to explain

Activity:

Part 1 Ideas: Students walk around the school with a sketchpad

Science/manufacturing industries: If the model involves the use of pulleys, gears or inclined planes

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 3 of 31

Content Teaching & Learning STEM Resources & Stimulus

similarity (ACMMG220)

recognise that if two figures are similar, they have the same shape but are not necessarily the same size (Reasoning)

match the sides and angles of similar figures

use the enlargement transformation and measurement to determine that the size of matching angles and the ratio of matching sides are preserved in similar figures

name the vertices in matching order when using the symbol ||| in a similarity statement

choose an appropriate scale in order to enlarge or reduce a diagram

looking for ways they would improve the physical environment. Big projects might include landscaping or construction; small projects might include comfort items or designing small decorative features.

Students make sketches of their ideas showing approximate location, shape and size. Some students might ‘tell a story’ about their idea.

Once back in class, students display their ideas and the class does a ‘walking tour’ around the room, browsing one another’s sketches.

Students then decide whether they want to continue with their own idea or adopt/adapt that of a classmate. Students may work as individuals or in small groups for the second part of the activity.

Part 2 – Model: Students make a rough model of their idea – at this stage the model is intended to help students give structure to their idea. Making the model will require students to think in more detail than was required for the sketch. At this stage the model is not to scale. (If possible, creation of the model is done in cooperation with other faculties to allow students access to the materials and facilities most suitable for the type of model they wish to make. This also allows students to engage teachers of their own choosing to develop the discussion of their idea beyond the maths classroom.)

Part 3 – Improve the model: Students work out the dimensions of the ‘real thing’. Students decide on the best way to record these and then decide whether their model needs adjustment. (Students do not need to re-make their models. The adjustments could be described in a diagram.)

Link to learning:

In creating the model, students have employed several mathematical processes, possibly without realising how complex these are:

the initial idea: symmetry, position, orientation, area, weight, height

students could involve their Science teachers to help build their knowledge and understanding of this component of the ‘Physical World’ from Stage 4 in the Science K-10 (http://syllabus.nesa.nsw.edu.au/science/science-k10/content/982/)

Technology/ICT: Students who have access to a digital printer may use this technology to create their model. Students who have ‘virtual world building’ skills may create their model within a game platform.

Engineering/construction/ manufacturing industries:

Students who enjoy working with materials may construct their model from wood, cloth, metal or plastic*.

Note to teachers: Opportunity for peer-to-peer coaching.

If students have used different methods of model-making they could describe/demonstrate/share those skills with classmates.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 4 of 31

Content Teaching & Learning STEM Resources & Stimulus

sketching the diagram: scale and proportion, angles, lengths

creating the model: scale, proportion, similarity

Teachers can present the learning goals for this unit as simply refining skills that students have already demonstrated. Improving the model will continue as a theme for the learning. The importance of an accurate model can be discussed in terms of its manufacture, costing, gaining approval or gathering student votes to have something built.

Consolidation for skill development:

The teacher explicitly teaches:

that if two figures are similar, they have the same shape but are not necessarily the same size.

that the model in its current form may be ‘similar’ in the normal use of the word, but to be mathematically similar all angles in the model must be the same size as they would be in the final product, and all lengths must have been changed by exactly the same scale factor.

students to recognise whether shapes are mathematically similar, including being able to: (Note to teachers: students are not required to use Similarity Tests as described in the 5.2 Outcomes.) o match the sides and angles of similar figures o use the correct convention for naming of vertices and edges o name the vertices in matching order when using the symbol ||| in

a similarity statement o use the enlargement transformation and measurement to

determine that the size of matching angles and the ratio of

matching sides are preserved in similar figures

Practice:

School-based and online worksheets could be used as resources

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 5 of 31

Content Teaching & Learning STEM Resources & Stimulus

enlarge diagrams such as cartoons and pictures (Reasoning)

construct scale drawings

investigate different methods for producing scale drawings, including the use of digital technologies (Communicating, Problem Solving)

determine the scale factor for pairs of similar polygons and circles

apply the scale factor to find unknown sides in similar triangles

calculate unknown sides in a pair of similar triangles using a proportion statement

apply the scale factor to find unknown lengths in similar figures in a variety of practical situations

Activity: Students practise scale drawings by a range of methods, such as:

1. Reduction and enlargement from a centre – real things (TES: https://www.tes.com/teaching-resource/reductions-and-enlargement-from-a-centre-real-things-11300111). Students use their own images to explore reduction and enlargement.

2. Reduction and enlargement of a cartoon using a grid. Students enlarge cartoon images (http://www.free-for-kids.com/drawing-grid-enlargement-worksheet.shtml)

3. Reduction and enlargement using technology: a. In word-processing software students insert an object and

enlarge and reduce it by ‘dragging’ corners. Teacher asks students to note the impact of dragging a side instead of a corner to emphasise that in Similar Figures there is enlargement/reduction without distortion.

b. Graphing software such as GeoGebra (Interactive example in Geogebratube: https://tube.geogebra.org/m/bQ8xwXPx?doneurl=%2Fsearch%2Fperform%2Fsearch%2Fenlarge)

c. Online applets such as ‘Playing with Dilations’ (National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities&from=category_g_4_t_3.html) and 'Similar Triangles' (Maths Open Reference: http://www.mathopenref.com/similartriangles.html)

d. Students create an artwork or design based on similar figures using the tool of their choice.

Link to learning:

Reduction and enlargement by hand requires students to look at an object or shape critically in order to select key points. This develops their awareness of vertices and edges. Manual measurements and repeated application of the scale factor reinforces that all edges are equally affected by scaling. The use of digital technologies allows

Students prepare for class by finding two diagrams or photos of something from their VET and/or career area of interest. One about the size of a postage stamp. One about, but less than, A5. eg: a piece of machinery, a tool, a farm animal or cropping plant, medical equipment, sports equipment, nail polish bottle, car, electric mixer.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 6 of 31

Content Teaching & Learning STEM Resources & Stimulus

students to efficiently ‘test the theory’ that angles and side ratios are preserved in similar figures.

Often, rather than needing to recreate an entire image we just need to know a side length. If similarity has been established, using the scale factor in a calculation is the most efficient method. Working with original images of the students’ choice clarifies the importance of matching edges. One could even use the scale factor determined by comparing two renderings of a bull’s leg (or many other features in an image) to redraw an enlargement of his face. This may be more obvious for students than comparing different edges of a triangle.

Consolidation for skill development:

The teacher explicitly teaches:

that scale factors are calculated from pairs of sides that have been correctly matched.

how to calculate the scale factor given that two images are similar.

that scale factor allows students to check their answers based on anticipated results: o Scale factor > 1 produces an enlargement o Scale factor < 1 produces a reduction o Scale factor = 1 produces a congruent figure (MA4-17MG

revision)

how to use a scale factor to find the length of an unknown side or radius

Practice:

School-based and online worksheets could be used as resources, such as: ‘Using Similar Polygons’ worksheet: (http://cdn.kutasoftware.com/Worksheets/Geo/7-Using%20Similar%20Polygons.pdf)

Improve their model made in the activity above o Noting again that the models are probably not mathematically

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 7 of 31

Content Teaching & Learning STEM Resources & Stimulus

similar, students determine the scale factor for two or more pairs of edges of their model.

o Students compare results and discuss how their models could be adjusted to make them more proportional.

interpret and use scales in photographs, plans and drawings found in the media and in other key learning areas

solve problems using ratio and scale factors in similar figures (ACMMG221)

Activity: Sometimes scale and proportion are deliberately misused to influence the viewer - eg caricatures. At other times, an incorrect scale or proportion would be disastrous - eg building a skyscraper. There are also times when a 1:1 scale is required.

View the video: Shaun Tang draws The Boy (https://www.youtube.com/watch?v=ZL4OYbAHuwQ). The education resource for The Boy this is at: (Australian Centre for the Moving Image - ACMI) https://www.acmi.net.au/education/shaun-tan-education-kit/the-boy/ o Discuss: Is this character to scale? How do you think this

character compares to a more ‘correct’ drawing of a boy?

View the video: Small Scale Measurements (Splash ABC: http://splash.abc.net.au/home#!/media/155036/) o Discuss: Think about all the things in the universe. Think about

the ones you could draw at their actual size? For everything else, you have to use scale.

o Sketch: an ant so you can see its features, the planet Saturn, a strawberry - actual size. For each, write whether the scale factor is > 1, < 1 or = 1.

View the video: Construction Project Coordinator (Ace Day Jobs – ABC: http://www.abc.net.au/acedayjobs/cooljobs/profiles/s2442978.htm) o Discuss: How do all the people on a building site know what to

do?

View the video: Simulation Coordinator (Ace Day Jobs – ABC: http://www.abc.net.au/acedayjobs/cooljobs/profiles/s2050243.htm) o Discuss: How important is it that the model and practice

Science/human services/ primary industries: Diagrams of living things are drawn to scale and the scale will vary depending on the purpose of the diagram. Students can easily explore and apply scale using a microscope and an object such as a sliver of tissue.

Students with an interest in health or sport sciences could explore a range of diagrams such as The Human Hand (Healthline: http://www.healthline.com/human-body-maps/hand),Teeth (Scootle: http://www.scootle.edu.au/ec/viewing/L730/index.html) and Hearing (Scootle: http://www.scootle.edu.au/ec/viewing/L721/index.html). Are these diagrams to scale? If not - why not?

Technology/entertainment industries: Computer-generated animation brings storytelling to life. This is demonstrated in ‘The Lost Thing’ trailer which builds upon Shaun Tans ‘The Boy’ (http://www.thelostthing.com/). The

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 8 of 31

Content Teaching & Learning STEM Resources & Stimulus

equipment is exactly to scale in this situation?

Link to learning:

By viewing common applications of scale, students realise they already use it on a daily basis. Students will also see that an understanding of scale is assumed in many workplaces and hence might value the learning more than if examples are limited to standard shapes in isolation.

Teachers could emphasise the value of a workforce that can read a plan correctly by showing this humorous video of the alternative: ‘... Road Construction Fails…’ (https://youtu.be/ODelzfAmauA)

Consolidation for skill development:

The teacher explicitly teaches:

how to locate the scale on a map or diagram

different presentations of scale and their interpretation

how to set up a system to solve a problem using ratio

how to set up an equation to solve a problem using scale factor

Practice:

School-based resources and online worksheets could be used such as:

‘Solving Ratio Word Problems’ – Thinking Blocks (http://www.thinkingblocks.com/thinkingblocks_ratios/tb_ratio_main.html). This interactive online resource steps students through a concrete method of setting out and solving a ratio problem. (Also available as an app)

Working with scale worksheet on a map (http://www.assessmentforlearning.edu.au/verve/_resources/Room_changes_working_with_scales.pdf) – This could be used as an Assessment for Learning (AFL) opportunity.

Improve the model o Students chose an appropriate scale to sketch a map of the

education resource: Shaun Tan’s The Lost Thing: From Book to Film (ACMI) is at: https://2015.acmi.net.au/media/428194/shaun-tan-ed-kit.pdf Note to teachers: a message about resilience for students is in this too - ask students to note how long the production process took.

Students with an interest in animation can delve further into character creation in the education resource for The Boy at: (Australian Centre for the Moving Image - ACMI) https://www.acmi.net.au/education/shaun-tan-education-kit/the-boy/

Engineering/construction trades: Building plans assume that people have a good understanding of scale.

Students with an interest in construction could explore the diversity of this industry through viewing videos such as ‘Construction of a 797 Dump Haul Truck’ (Education Services Australia: http://www.oresomeresources.com/media_centre_view/resource/video_797_dump_haul_truck_construction/category/mining_videos/section/media/parent/) and ‘Rectangular Stadium’ (Splash ABC: http://splash.abc.net.au/home#!/medi

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 9 of 31

Content Teaching & Learning STEM Resources & Stimulus

area that they would like to improve with their idea and locate their idea on the map to show its relative size.

a/1454941/)

find perimeters of parallelograms, trapeziums, rhombuses and kites (ACMMG196)

investigate the relationship between features of circles, such as the circumference, radius and diameter; use formulas to solve problems involving circumference (ACMMG197)

find the perimeters of quadrants and semicircles

find the perimeters of simple composite figures consisting of two shapes, including quadrants and semicircles

Knowledge & Understanding Review: Vocabulary of shape

For teachers: Students should be able to communicate using the following language: centimetre, dimensions, distance, estimate, kilometre, length, measure, measuring device, metre, millimetre, perimeter, ruler, tape measure, trundle wheel, width.

'Perimeter' is derived from the Greek words that mean to measure around the outside: peri, meaning 'around', and metron, meaning 'measure'

“The names for some parts of the circle (centre, circumference, diameter, quadrant, radius, sector and semicircle) are introduced in Stage 3. The terms 'arc', 'tangent', 'chord' and 'segment' are introduced in Stage 4.

Some students may find the use of the terms 'length/long', 'breadth/broad', 'width/wide' and 'height/high' difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, when describing diagrams. Students should be encouraged to speak about, listen to, read about and write about the dimensions of given shapes using various combinations of these words, eg 'The length of this rectangle is 7 metres and the width is 4 metres', 'The rectangle is 7 metres long and 4 metres wide'. Students may also benefit from drawing and labelling a shape given a description of its features in words, eg 'The base of an isosceles triangle is 6 metres long and its perimeter is 20 metres. Draw the triangle and mark on it the lengths of the three sides'.

In Stage 3, students were introduced to the term 'dimensions' to describe the length and width of a rectangle. However, some students may need to be reminded of this.”

Engineering/construction/ primary industries: “The ability to determine the perimeters of two-dimensional shapes is of fundamental importance in many everyday situations, such as framing a picture, furnishing a room, fencing a garden or a yard, and measuring land for farming or building construction.

Students should develop a sense of the levels of accuracy that are appropriate to a particular situation, eg the length of a bridge may be measured in metres to estimate a quantity of paint needed, but would need to be measured much more accurately for engineering work.”

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 10 of 31

Content Teaching & Learning STEM Resources & Stimulus

use Pythagoras' theorem to find the length of an unknown side in a right-angled triangle

write answers to a specified or sensible level of accuracy, using an 'approximately equals' sign, ie ≅ or ≈

solve a variety of practical problems involving Pythagoras' theorem, approximating the answer as a decimal

apply the scale factor to find lengths in the environment where it is impractical to measure directly, eg heights of trees, buildings (Problem Solving)

apply the scale factor to find unknown lengths in similar figures in a variety of practical situations

Activity: Measure the inaccessible.

Students visit a location that provides a variety of inaccessible lengths. For example: a body of water, a bridge or a tall building.

In small groups or pairs, students gather information and record it on diagrams to later calculate unknown lengths using:

1. Pythagoras’ theorem 2. Similar triangles 3. Trigonometry

If moving outside the classroom is not possible, an alternate approach is to consider inaccessible lengths due to solid objects, including distance.

Visit the location and describe the objective to students. Ideally they would be shown a photograph of the area to allow them to participate in planning. Inform students that they will be ‘building’ right-angled triangles on the unmeasurable length so that two sides or one side and an acute angle are known. Samples of field drawings for trigonometry, similarity and Pythagoras' theorem in the real world to share with students can be found at: https://www.tes.com/teaching-resource/samples-of-field-drawings-for-trigonometry-similarity-and-pythagoras-theorem-in-the-real-world-11464914. Note to teachers: students can know what measurements are required without yet having to understand how they will be used.

Equipment: Appropriate measuring and recording tools including a method to measure angles in the environment.

On location: Students sketch maps or diagrams, take appropriate measurements and show and label these on their sketches. Students estimate the lengths that they will later calculate.

Link to learning:

This activity shifts Pythagoras’ theorem, similar triangles and trigonometry from being three separate topics, to three useful methods

Science/tourism industries: Observatories, skywalks, treetop walks and other viewing platforms are important to many varieties of tourist destinations, particularly those tourists who are interested in the environment. The design and maintenance of these uses the mathematics of measuring the inaccessible. Students could discuss what they consider the world’s most famous lookout point.

Technology/electrical trades: Video – high power line workers (https://youtu.be/r_1T2_l43Xo). Students note the electrician’s explanation towards the end of the video about the safety of the job – everything is perfectly planned. Essential to that planning is the mathematics of measuring the inaccessible that guides the helicopter pilot’s approach and angle of hover.

Engineering & Technology/construction industries: Video – Demolishing the Old Bay Bridge East Span (https://www.youtube.com/watch?v=9xhcTDLM4Ss). Students note the use of triangles in the construction. This is because of the strength of a triangle. The building of the bridge

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 11 of 31

Content Teaching & Learning STEM Resources & Stimulus

of approaching the same practical problem.

Stage 5.1 students are at a different location in learning for each of these methods:

Pythagoras theorem: recall past learning

Similar triangles: apply recent learning

Trigonometry: context for new learning

Diagrams could be displayed in the classroom grouped by method and referred to throughout this section of the program.

Consolidation for skill development and practice:

On their return to classroom, students:

apply Pythagoras theorem to calculate inaccessible lengths of triangles where two sides are known. Compare to estimations. o Explicit revision of Pythagoras theorem as required. o Discussion of appropriate accuracy and hence rounding of

calculated answers. Correct use of units.

apply scale factor to calculate inaccessible lengths of triangles where matching lengths in a similar triangle are known. Compare to estimations. o Deliberate recall and application of recent learning o Discussion of appropriate accuracy and hence rounding of

calculated answers. Correct use of units.

examine the diagrams where one side and one angle of the triangle were measured. Clarify that these have insufficient information for use with Pythagoras’ theorem or similarity. However, another method exists – Trigonometry – which is new learning.

and now its demolition rely heavily on Pythagoras’ theorem and similar triangles. Students will be able to see that the original bridge builders had a completely inaccessible distance to measure high above the water. Teachers can point out that the students’ own calculators are more powerful than the tools those builders had.

find examples of similar figures embedded in designs from many cultures and historical periods

Developing understanding: Why all this emphasis on triangles?

Video – Pythagorean Triangles (Splash ABC: http://splash.abc.net.au/home#!/media/1469315/) – ancient

STEM: Building with Pasta (NASA: https://www.nasa.gov/offices/education/programs/national/summer/education_resources/engineering_grades7

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 12 of 31

Content Teaching & Learning STEM Resources & Stimulus

(Reasoning) Egyptians realising that a triangle with sides of 3, 4 and 5 would always be a perfect right angle. o Students can check this for themselves with precisely cut

matchsticks or straws

Video – Exploring Triangles (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/triangles/trianglesSmall.html) – points out triangles in buildings o Students count any triangles they can see in their immediate

environment and consider their role.

Video – The Complex geometry of Islamic Design (TedEd: http://ed.ted.com/lessons/the-complex-geometry-of-islamic-design-eric-broug) – circles, triangles and stars in design o Students can search online for other strong geometrical

traditions in the designs from other regions of the world

Interactive slideshow – Measuring with Triangles (Scootle: http://www.scootle.edu.au/ec/viewing/L2326/index.html) – Ancient Greek use of similar triangles and shadows. reviews similarity and demonstrates other useful triangle facts. o Students engage with the interactive questions

Video – Just Another Day at the Office (Volvo Ocean Race: https://www.youtube.com/watch?v=aN4CBbeSbP4&feature=youtu.be) – does not reference navigation directly but provides an interesting point to start the discussion about trigonometry in navigation.

-9/E_spaghetti-anyone.html#.V2iVwnozcvk) – demonstrates to students the strength of a triangle compared to other shapes.

Science/tourism/primary industries: While GPS technology is available almost everywhere, the ability to locate yourself on a map is a vital skill for industries that operate in isolated areas. Triangulation will still work when your batteries are flat or you cannot locate a satellite.

Navigation Expert Advice (REI: https://www.rei.com/video/2010/10/28/navigation-expert-advicel-triangulation.html) – demonstration of using triangulation with a map to make sure you don’t get lost in the wilderness.

o Students can test this for themselves using a local map and compass

Mapping Farmland using Area and Trigonometry (Splash ABC: http://splash.abc.net.au/home#!/media/86350/mapping-farmland-using-area-and-trigonometry) – technology is changing mapping in remote areas

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 13 of 31

Content Teaching & Learning STEM Resources & Stimulus

establish the relationship between the lengths of the sides of a right-angled triangle in practical ways, including with the use of digital technologies

use dynamic geometry software to investigate the properties of similar figures (Problem Solving)

use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (ACMMG223)

identify the hypotenuse, adjacent sides and opposite sides with respect to a given angle in a right-angled triangle in any orientation

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

select and use appropriate trigonometric ratios in right-

Activity: Who needs to understand triangles? Show the video: ‘World’s Largest Urban Zipline’: (https://www.youtube.com/watch?v=YcwrRA2BIlw&feature=youtu.be)

Concrete activity: o Equipment: straws or matchsticks cut to different lengths, rulers,

protractors and worksheet. 1. Students select three lengths at random and decide whether they

can form an acute triangle. a. If a triangle can be formed, students test whether the same three

lengths can form any other acute triangles - rotations and reflections are not new triangles.

b. Students may begin to develop an understanding of what is required of lengths for them to form a triangle.

2. Students select two lengths at random and form a right-angle. a. Students measure the missing side length needed to form a

triangle (the hypotenuse) and decide whether any other third side is possible.

b. Students use a protractor to measure the two acute angles and decide whether any other angles are possible.

c. Students may begin to develop understanding of the hypotenuse as the longest side and that the longest side is always opposite the right angle.

Digital activity: Similar Triangles and Dynamic Software (https://www.tes.com/teaching-resource/properties-of-similar-figures-and-dynamic-software-11305724)

Differentiation – extend: Part 4 of this Activity is designed for students who have demonstrated a desire to understand where mathematical processes come from. The majority of Stage 5.1 students may be best served by simply presenting the trig ratios and calculator functions as a set of steps to be followed.

3. Similar Triangles and introduction to Trigonometry worksheet (https://www.tes.com/teaching-resource/similar-triangles-and-

Science/tourism industries: Students search online for amazing zipline attractions – longest, fastest, most frightening, most eco, most urban. Teacher can then explain that the planning and erection of these is completely reliant on trigonometry. Students consider the consequences of getting those calculations wrong.

Technology/electrical trades: Students interested in technology can consider the amount of technology involved in communicating with and tracking large passenger jets as they approach airports. Much of that communication and tracking is to make sure the plane is sticking to a flight path that has been calculated using trigonometry. Landing a plane video (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).

Engineering/construction industries: Photo challenge. Students look for triangles in the built environment and photograph these. Create a slideshow to share with the class.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 14 of 31

Content Teaching & Learning STEM Resources & Stimulus

angled triangles to find unknown angles correct to the nearest degree

introduction-to-trigonometry-11304321) a. Students prove a set of right-angled triangles is similar by

measuring angles from a printed diagram. Use given side lengths to calculate scale factor and hence calculate missing side length

b. Students complete a table of opposite, adjacent and hypotenuse side lengths for a set of similar triangles then calculate the ratios O

H,A

H,O

A for each triangle

c. Students are shown right angled triangles whose hypotenuse connects the centre of the unit circle to a point on its perimeter. They realise that the hypotenuse length is 1 unit. They then consider whether the diagram can be used to find height and width of each triangle.

d. Students add their own triangles to the diagram and hence consider that a triangle could be drawn for every angle between 0° and 90° degrees

Link to learning:

Discuss:

There is nothing ‘random’ about right-angled triangles. Once one acute angle and one side are described, the remaining angle and sides are completely determined.

(differentiated) Every right-angled triangle with an acute angle of 𝑥° is similar to every other right-angled triangle with an angle of 𝑥°

(differentiated) Students move towards understanding of constancy of sin, cos & tan ratios, having performed the calculations themselves in the worksheet

(differentiated) Trigonometric ratios (SOHCAHTOA) are simply a ‘short cut’ for solving sides and angles using the recorded lengths of the triangles with hypotenuse length one (from the unit circle) and similarity.

Consolidation for skill development:

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 15 of 31

Content Teaching & Learning STEM Resources & Stimulus

The teacher explicitly teaches:

how to identify the hypotenuse, adjacent sides and opposite sides with respect to a given angle in a right-angled triangle

how to define the sine, cosine and tangent ratios for angles in right-angled triangles

how to identify what is known and what is required in a problem

how to select the most useful trigonometric ratio (SOHCAHTOA)

how to substitute known values and solve

consideration of units

calculator use

Practice:

Apply trigonometry to the last of the inaccessible lengths in the class’ sketches.

Use online interactives such as: o Labelling sides and setting up SOHCAHTOA (Scootle:

http://www.scootle.edu.au/ec/viewing/L6561/asset1.html) o Exploring Trigonometric Ratios (Learn Alberta:

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/object_interactives/trigonometry/use_it.html)

o Differentiation – extend: Trigonometry (Splash ABC: http://splash.abc.net.au/home#!/topic/496568/trigonometry/interactive) – ‘games’ that demonstrate the trigonometry ratios in problem solving and each time relate the triangle back to the unit circle and hence similarity

Angles of Elevation and Depression Online lesson – sections 1–5. (MyMaths.co.uk: Angles of Elevation and Depression: http://www.mymaths.co.uk/samples/sampleLessonElevation.swf) o 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.

Further trigonometry information and interactive practice (Australian

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 16 of 31

Content Teaching & Learning STEM Resources & Stimulus

Mathematical Sciences Institute (AMSI): http://www.amsi.org.au/ESA_middle_years/Year9/Year9_md/Year9_2c.html#intro)

solve right-angled triangle problems, including those involving angles of elevation and depression (ACMMG245)

connect the alternate angles formed when parallel lines are cut by a transversal with angles of elevation and depression (Reasoning)

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 aims to introduce the terminology without affecting tentative confidence with recent trigonometry learning.

Activity: Practical situations and trigonometric diagrams

Students work in groups of four.

Requirements: o An environment where students can be safely situated at

different heights, eg: staircase, sloping landscape, playground equipment, upper floor windows.

o One clipboard per group. o One measuring tape or trundle wheel per group. o Students willing to work with estimated angles

Objective – calculate a distance that would be difficult to measure by gathering sufficient information for one of the trigonometric ratios. o Two group members (objects) position themselves at different

heights. The other members (surveyors) sketch the situation, estimate angles and measure one accessible distance.

o Instruction to ‘objects’ once 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 take one accessible measurement.

Science/tourism industries: Ecotourism takes visitors into the environment and often seeks out unique vantage points from which to capture a view. Students could investigate:

the relationship between elevation and distance to the horizon why Uluru is not climbed by as many visitors these days despite the view from the top the range of motion of the human neck and comfortable angles of elevation compared to straining ones.

Engineering/primary industries: Irrigation is costly and crucial in Australian farming. For water to move efficiently across the landscape farmers need to work with angles of elevation and depression.

Engineering/Retail industries: Angles of elevation and depression describe our line of sight. While the retailer designing a window display may not actually use sin, cos or tan, their display will have a greater impact if they consider the angle of

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 17 of 31

Content Teaching & Learning STEM Resources & Stimulus

Angles of Elevation and Depression: in the field and in the classroom activity (https://www.tes.com/teaching-resource/angles-of-elevation-and-depression-in-the-field-and-in-the-classroom-11305036)

In the classroom: o Redraw field sketches as diagrams for calculations. o Discuss the limitations on accuracy and possible improvements.

Link to learning:

Discuss:

How can we describe the angles that were estimated in 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 reminded of any previous maths topics about angles? (Angles in parallel lines.)

Consolidation for skill development:

The teacher explicitly teaches:

how to identify angles of elevation and depression

how to solve a variety of practical problems, including those involving angles of elevation and depression, when given a diagram

Practice:

Improve the model: Students draw an elevation sketch showing the location of their idea and the best place from which it will be viewed or angles of approach, labelling these as angles of elevation or depression.

Final evaluation of the model: o Students consider their different drafts of the model of their

elevation from the point of view of potential customers. Developing trigonomentric skills in the classroom can increase students’ ability to estimate and interpret angles, distances and their relationship to one another.

Understanding elevation and angles is also important in the retail sector from an OHS perspective. Students could look for regulations in articles such as Ladders - What are the Rules and Regulations? (http://www.ohsrep.org.au/faqs/ohs-reps-@-work-other-/ladders-what-are-the-rules-and-regulations)

Engineering/automotive industries: Approach and departure angles limit the steepness of a ramp or terrain that a vehicle can handle before ‘bottoming out’ or becoming stuck. Students could investigate how these vary between different types of vehicles. Then, scroll to ‘Measuring Your Angles’ in the article ‘How Off-roading Works’ (How Stuff Works: http://adventure.howstuffworks.com/outdoor-activities/off-roading/off-roading1.htm) and relate that description to their angles of elevation and depression activity.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 18 of 31

Content Teaching & Learning STEM Resources & Stimulus

initial idea. One or two homework sessions can be devoted to producing their final copy for display.

o Using red, yellow and green sticker dots (or similar) students evaluate one another’s models in terms of what chance they think the idea has of being adopted by the school’s leadership team. Red - none, yellow - worth trying, green - a good chance. As part of their evaluation students should consider whether the model is clear and easy to interpret.

Note to teachers: students are commenting on whether they think the school leadership team would accept an idea – not whether they themselves like the idea. This makes this evaluation of the models more ‘friendly’.

calculate the areas of composite shapes (ACMMG216)

calculate the areas of composite figures by dissection into triangles, special quadrilaterals, quadrants, semicircles and sectors

calculate the area of an annulus (Problem Solving)

Activity: Wrap it

Equipment: prizes, wrapping paper (recycled newspaper), items in various shapes – some simple to wrap and some difficult.

Students are given an object at random and some wrapping paper.

They race to be first to wrap their object and produced the best wrapped object.

There is a prize.

The value in this activity relies on students noticing that the race is unfair due to the difference in items to be wrapped. Teachers can respond to this injustice by providing additional prizes to any students who can use a mathematical term in an explanation of why they thought themselves disadvantaged.

Teachers can direct the discussion of fairness towards simplicity of shape.

Link to learning:

Are all plane shapes combinations of other plane shapes?

Are all areas of plane shapes combinations of other areas of plane

STE(A)M: Present the artwork of Christo and Jeanne-Claude (http://christojeanneclaude.net/artworks/realized-projects) by sharing the galleries on their website.

Discuss: Do students think these artists calculate the amount of material they will need before commencing or just turn up with as much as they can carry? How would such calculations be performed?

Students browse the site for themselves and choose the projects they think would have been the simplest and the most difficult.

Discuss the shapes that students can see. Does the wrapping make shapes easier or harder to see

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 19 of 31

Content Teaching & Learning STEM Resources & Stimulus

shapes?

What then would be the two or three most essential plane shape area formulae to memorise so that all other areas of plane shapes could be calculated?

Consolidation for skill development and practice:

The teacher explicitly teaches:

Formula and diagrams for area of: o Circle

Hence annulus, semicircle, quadrant and sector Video – Area of a Circle, How to get the Formula

(mathematicsonline: https://youtu.be/YokKp3pwVFc) o Rectangle

Hence triangle and special quadrilaterals Problem Solving - students cut shapes to discover the

rectangular area formula within the formulae for triangles and other quadrilaterals (Area problem solving samples: https://www.tes.com/teaching-resource/area-of-rectangles-connected-to-triangles-and-special-quadrilaterals-11307354) – each formula can be related to Area = base x height

Online interactives:

Area (BBC Bitesize: http://www.bbc.co.uk/bitesize/ks3/maths/measures/area/activity/) – simple rectangle and triangle revised and then applied to a complex shape showing adding and subtracting areas.

Compound shapes (Scootle: http://www.scootle.edu.au/ec/viewing/L153/index.html) – shapes on a grid, area calculated by counting squares – differentiated

Annulus (Maths is Fun: https://www.mathsisfun.com/geometry/annulus.html) – examples

compared to the original object?

Science/Technology: Do students know much about fabric strength versus flexibility versus weight? What properties would a fabric need to be stretched over a tree? What would be the challenges? Would the same fabric be the best thing to cover the Opera House? Students could engage their Tech teachers in a discussion about the role of fabrics in a range of industries. Eg: waterproof and chemical-proof gowns in hairdressing salons, fireproof suits in auto racing, reflective fabrics in construction, breathable fabrics in sport, etc.

Engineering: Problem Solving: students consider how the artists actually get the fabric up and over the large and difficult-to-access objects. What tools or equipment would they need? If resources allow, the class could attempt to wrap an awkward and large object within the school.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 20 of 31

Content Teaching & Learning STEM Resources & Stimulus

and questions

Sector applet (Maths Open Reference: http://www.mathopenref.com/arcsectorarea.html)

use ratios to compare quantities measured in the same units

solve a variety of practical problems involving the areas of quadrilaterals and composite shapes

apply properties of geometrical shapes to assist in finding areas, eg symmetry (Problem Solving, Reasoning)

Activity: Shape-recognition walk

Class takes a walking tour of the school or a nearby locale to sketch the environment but with everything shown as a composite shape. A 2D sketch is required – let students know this in advance.

Instruct students to also look for symmetry in built and natural objects and highlight this in the sketch

Introduction video ‘Shapes glorious shapes’ (ABC Splash: http://splash.abc.net.au/home#!/media/1566372/shapes-glorious-shapes) can assist students recognition of shape in buildings o Alternative if a walking tour is not possible:

sketch from projected images sketch on top of printed images use drawing software with a digital image

Link to learning:

The STEM stimulus, consolidation and practice here focus on careers where recognition of shape is essential. Teachers could replay the aircraft carrier video at 2:40 where the pilot’s point of view of the ship is demonstrated with a piece of paper and discuss: what it means when the deck is looking like a short rectangle compared to a long one - students could model this for themselves.

Consolidation for skill development and practice:

Renovate, Calculate! (Scootle. DET NSW): http://www.resources.det.nsw.edu.au/Resource/Access/2bb8fd0c-1f62-4658-be9a-6fdae3817fd3/1) o A self-contained sequence of engaging videos, quick maths

quizzes and learning in the context of renovating a garage

STEM/VET: Students view the video Carrier Pitching Flight Deck - Landing and Take offs with Bad weather Condition (https://www.youtube.com/watch?v=MI7cYywK-fg). This video is 16 mins long. Students could note each job they hear named and then tick every time they hear a maths reference. These will be about pitch, angle, time, fuel, graphs, charts, pattern, altitude, wave size. When video concludes students discuss the different skilled trades that are required to run a ship and what geometry they need to understand. Eg: plumbers need to know pipe diameters, water volume, pressure and angles; landing signal officers need to have strong angle and distance and speed estimation skills, glaziers, painters, metal workers, electricians, cooks, healthcare workers, communications technicians, all have mathematical calculations to make as part of their daily responsibilities.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 21 of 31

Content Teaching & Learning STEM Resources & Stimulus

space to make a studio. o For each of painting, metalwork and carpentry students view a

young person beginning in the trade. Mathematics is described in context.

o An interactive learning activity relevant to each trade first teaches and then tests students.

o Students will require guidance to navigate the sequence of activities - returning to the home page is required for progression between themes.

Differentiation – Extend: For students interested in digital imagery and computer generated images (CGI).

Video – How Benjamin Button got his Face (TED Talk: http://ed.ted.com/lessons/how-benjamin-button-got-his-face-ed-ulbrich#watch)

solve problems involving the surface areas of right prisms (ACMMG218)

identify the edge lengths and the areas making up the 'surface area' of rectangular and triangular prisms

visualise and name a right prism given its net

visualise and sketch the nets of right prisms

find the surface areas of rectangular and triangular prisms, given their net

Activity: Foil prank

This activity may require two lessons depending on lesson length so that students have time to perform the calculations and receive assistance as required

Preparation: seek approval!

Equipment: rolls of aluminium foil, measuring tape, scissors

Students: o search ‘foil prank’ images for inspiration

(https://www.google.com.au/search?q=foil+prank&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjUl8uB1snNAhUEupQKHUa-D4oQsAQIHA&biw=940&bih=475)

o calculate what area of foil they have in the available rolls o in small groups, find, sketch and measure an object to be

wrapped within the school grounds - the sketch should use the same technique as the ‘shape-recognition walk’ and represent the object as a combination of basic shapes

Science: NASA uses gold foil to wrap satellites and line astronauts’ visors. Students can read ‘Why gold foil is valuable to space exploration’ (http://www.geek.com/science/geek-answers-why-does-nasa-use-so-much-gold-foil-1568610/). Understanding the price of gold foil will emphasise the need to make accurate surface area calculations to avoid wastage.

Science/hospitality industries: Foil is among many of the food-packaging options. Students can read about Food Packaging - Roles, Materials and Environmental Issues (http://www.ift.org/knowledge-

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 22 of 31

Content Teaching & Learning STEM Resources & Stimulus

o return to classroom to calculate the total surface area of the chosen object and decide whether there is sufficient foil by: redrawing the object showing the basic shapes being used

for each of its individual faces - approximation is encouraged writing the measurements onto the shapes naming each shape and writing its area formula

o present sketch and calculations to the class o class checks calculations and votes on which project/s will be

wrapped (with permission)

Link to learning:

In order to calculate the total surface area of their object, students had to identify edge lengths and decide on shapes that approximated the object’s faces. Students have calculated the area of a range of shapes and checked the diagrams and calculations of classmates. This prepares them to work with nets of right prisms.

Consolidation for skill development:

The teacher explicitly teaches:

Naming right prisms

Recognising two end shapes and rectangles o present this as method for sketching a net given a right prism

Recognising nets for right prisms o recognising variations of nets – variations of cuboid shown here

(BBC Bitesize: http://www.bbc.co.uk/bitesize/ks3/maths/shape_space/3d_shapes/revision/3/ variations of cuboid shown here)

Systematic method for calculating total surface area given a net

Practice:

Make prisms by folding nets (printable nets on Scootle: https://www.tes.com/teaching-resource/assorted-nets-6263628) (more printable nets on mathworksheets4kids.com:

center/read-ift-publications/science-reports/scientific-status-summaries/food-packaging.aspx)

Technology/hairdressing industries: Hairdressers use aluminium foil to control colour. Students can read about the use of foil in salons (https://bellatory.com/hair/Hairdressing-How-To-Putting-Foils-in-Hair---Tips--Tricks--Advice-and-Know-How-for-Colouring-Hair-with-Foils). Discuss how hair length and volume dictate the total surface area (TSA) of foil and amount of product that will be required for a person’s treatment and hence how these must be considered in costing.

Technology/automotive industries: Cars can be wrapped in printed foils. Students can read about vehicle wrapping (http://solutions.3m.com/wps/portal/3M/en_EU/3MGraphics/GraphicSolutions/Applications/CarWrapping/).

Engineering/construction industries: Insulation materials include reflecting foil products and a lack of understanding about this material resulted in much-publicised deaths in Australia around 2010. Students can read about this at Product Safety Australia

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 23 of 31

Content Teaching & Learning STEM Resources & Stimulus

http://www.mathworksheets4kids.com/solid-shapes.php)

Fluency questions: school-based and online worksheets as resources, such as: o 3D shapes test (BBC Bitesize:

http://www.bbc.co.uk/bitesize/quiz/q44861627) o Surface Area using Nets (Khan Academy:

https://www.khanacademy.org/math/basic-geo/basic-geo-volume-surface-area/basic-geo-surface-area/e/surface-area)

(https://www.productsafety.gov.au/content/index.phtml/itemId/974027).

calculate the surface areas of rectangular and triangular prisms

apply Pythagoras' theorem to assist with finding the surface areas of triangular prisms (Problem Solving)

solve a variety of practical problems involving the surface areas of rectangular and triangular prisms

Activity: How many calculations do you really need to perform?

Equipment: prism blocks (empty boxes) and wrapping paper (newspaper)

Instruction: wrap the block (box) without any overlap

Teachers actively observe methods chosen by students. 1. Did any students cut out the net? 2. Did any students cut two end shapes and a separate rectangle for

each side? 3. Did any students cut two end shapes and one rectangle to wrap

around the ‘body’? 4. Did any students use a traditional gift-wrapping method?

Utilising observations:

1. These students are demonstrating application of recent learning and have shown ability to see the surface as a whole and plan before cutting.

2. These students show they can break a larger problem into its components.

3. These students are demonstrating problem solving skills, using an efficient, real-world, method.

4. These students have not interpreted the instructions correctly and require a second opportunity to complete the activity.

Link to learning:

Science/human services/primary industries/automotive industries: In each of these industries the use of chemicals and high temperatures means a risk of injury including burns. The severity of a burn is measured in percentage of a person’s total surface area that has been affected. The ‘rule of 9s’ is used to allow quick estimates of this so that emergency medical workers can triage appropriately. Eg, if four workers are burnt, the one with the highest percentage of total surface area will be treated as the greatest emergency. Students can see an example of the tables and diagrams used to help this calculation CHEMM (Chemical Hazards Emergency Medical Management) - Burn Triage and Treatment (https://chemm.nlm.nih.gov/burns.htm)

Activities:

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 24 of 31

Content Teaching & Learning STEM Resources & Stimulus

Previously students calculated total surface area given a net. Now they will calculate given a representation of the 3D prism in diagram or words. The following activity is designed to help students see that they have some choice in method and can reduce the total number of calculations by recognising the key features of prisms - identical end shapes and ‘the body’ joining them.

Consolidation for skill development:

Surface area worksheet (TES: https://www.tes.com/teaching-resource/area-and-surface-area-of-prisms-11315337) to be done with teacher direction.

Discuss: o Will there always be ‘one wrap-around rectangle’ on a prism?

If so, is this a useful test for a prism? Does it provide a useful way to draw a net? How are the length and width of that one rectangle found?

o Will there always be two identical ends on a prism? If so, is this a useful test for a prism? Does it provide a useful way to draw a net? Do you have to calculate both areas separately or is there a

short cut?

Practice: skills and drills

School-based resources and online worksheets could be used such as:

Thatquiz.org allows teachers to set up quizzes and share them to students via a URL. An example of such a Surface Area quiz is at: (Thatquiz.org) https://www.thatquiz.org/tq-4/?-j4080-l5-p0) which has been set for students to compare the surface area of two prisms. Difficulty level and question type can be adjusted to allow differentiated practice.

Surface area worksheets (Math-aids.com: http://www.math-aids.com/Geometry/Volume/) – include pyramids and cylinders. They can be adapted to identify prisms first, then calculate their

1. Students test whether these estimations are valid by representing their own bodies as a set of rectangular prisms and calculating their own total surface area then comparing to the percentages in the diagram for different parts of the body.

2. Role-play an industrial emergency and first aid including 000 calls and communicating about the percentage body surface area affected by burn.

3. Consult with their Science teachers about burn procedures in the school labs.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 25 of 31

Content Teaching & Learning STEM Resources & Stimulus

surface area.

Bringing it all together:

Pythagoras, Trigonometry, Similarity, Scale, Area and Total Surface Area. Through this program students have seen them used individually or together in design, manufacturing, building, modelling, forecasting, measuring, testing, planning, air travel, sea travel, medicine, art, food preparation, hairdressing, automotive industries, agriculture, space exploration, military operations, tourism and adventure travel, power supply and off-roading.

The 1998 Sydney Hobart Yacht Race film, when watched with a renewed awareness of mathematics in the world, is rich in STEM applications.

Triangles in yacht and sail design

Similar triangles created when sails are reefed in storm preparation.

Trigonometry in race planning and then in rescue missions

Area of sails on super-maxis compared to smaller boats

Total surface area of hulls and boat speed

Annulus shape of life raft design

Scale and map interpretation against storm forecasts in the decision to turn back or go forward

Angle of approach for rescue helicopters

Wave height and steepness compared to boat length

Assessment As Learning (AAL): Investigation - individual or group

Students select one of the following to research and answer using mathematical terms, diagrams and sample calculations connected to this T&L Program.

Racing yachts (individual) o Why do yachts have triangular sails?

STEM:

Video: 1998 Sydney Hobart Yacht Race film part 1 (https://youtu.be/wgsp_kHicu8)

(Teacher note: Part 2 follows as a second video - teachers should preview Part 2 as the loss of sailor’s lives is described there.)

This extraordinary event is a significant moment in Australia’s maritime history - and a feat in Science, Technology, Engineering and Mathematics. This might be watched in cooperation with the HSIE or Science faculty.

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 26 of 31

Content Teaching & Learning STEM Resources & Stimulus

o Explain why a ‘reefed’ sail is mathematically similar to the full sail.

o Explain when and why sails are reefed. o Find the measurements needed to draw a scale diagram of the

main sail of a super-maxi yacht beside a diagram of yourself drawn to the same scale

o Find the mast height of a super-maxi yacht and calculate how far a person at the top of the mast can see

Design a dome (group) o The fully described activity is at: TryEngineering.com:

http://www.tryengineering.org/lessons/designadome.pdf

Design a playhouse (group) o View video ‘Cardboard Playhouse’ (Design Squad Global:

https://youtu.be/O8AMJkqo2lw) o Students create their own design and present

scale drawing surface area calculation for sections that might be decorated trigonometry calculation to show angle of one element such

as slide, ramp or roof

Other investigations of the students’ choice connected to their area of interest and associated with a VET course of study

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 27 of 31

Assessment strategies

Worksheets

Topic Tests: Short answer and multiple-choice tests on accrued skills as the program is progressed through.

Working with scale worksheet on a map (http://www.assessmentforlearning.edu.au/verve/_resources/Room_changes_working_with_scales.pdf)

Student self-evaluation: Students rate their own development through this unit - their understanding and skills, their application to learning and

working mathematically. Students discuss these with one another and then with their teacher 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

The following URLs are correct at the time of publication

Additional Teacher Resources URLs:

The “8 Ways of Learning for Aboriginal students”: 8ways.wikispaces.com

‘Scale Drawing and Similarity AMSI: http://www.amsi.org.au/teacher_modules/Scale_drawings_and_similarity.html

Ratios and Rates AMSI: http://www.amsi.org.au/teacher_modules/rates_and_ratio.html

Surface Area and Volume of Prisms and Cylinders AMSI: http://www.amsi.org.au/ESA_middle_years/Year9/Year9_md/Year9_2a.html#intro

Teaching & Learning URLs of linked resources:

Enlargement/reduction from a centre – real things: https://www.tes.com/teaching-resource/reductions-and-enlargement-from-a-centre-real-things-11300111

Free enlargement using grid worksheets, Free-For-Kids.com: http://www.free-for-kids.com/drawing-grid-enlargement-worksheet.shtml

Geogebratube: https://tube.geogebra.org/m/bQ8xwXPx?doneurl=%2Fsearch%2Fperform%2Fsearch%2Fenlarge

National Library of Virtual Manipulatives Playing with Dilations: http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities&from=category_g_4_t_3.html

Maths Open Reference: http://www.mathopenref.com/similartriangles.html

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 28 of 31

Resources overview

Using Similar Polygons worksheet (http://cdn.kutasoftware.com/Worksheets/Geo/7-Using%20Similar%20Polygons.pdf)

Shaun Tan draws The Boy: https://www.youtube.com/watch?v=ZL4OYbAHuwQ

Education Resource for The Boy.(ACMI): https://www.acmi.net.au/education/shaun-tan-education-kit/the-boy/

Small Scale Measurements (Splash ABC): http://splash.abc.net.au/home#!/media/155036/

Construction Project Coordinator (Ace Day Jobs - ABC): http://www.abc.net.au/acedayjobs/cooljobs/profiles/s2442978.htm

Simulation Coordinator (Ace Day Jobs - ABC): http://www.abc.net.au/acedayjobs/cooljobs/profiles/s2050243.htm

‘... Road Construction Fails…’: https://youtu.be/ODelzfAmauA

‘Solving Ratio Word Problems’ - Thinking Blocks: http://www.thinkingblocks.com/thinkingblocks_ratios/tb_ratio_main.html

Working with scale worksheet on a map (Assessment for Learning) (http://www.assessmentforlearning.edu.au/verve/_resources/Room_changes_working_with_scales.pdf)

Samples of field drawings for trigonometry, similarity and Pythagoras' theorem in the real world (TES): https://www.tes.com/teaching-resource/samples-of-field-drawings-for-trigonometry-similarity-and-pythagoras-theorem-in-the-real-world-11464914

Pythagorean Triangles (Splash ABC): http://splash.abc.net.au/home#!/media/1469315/

Exploring Triangles (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/triangles/trianglesSmall.html

The Complex geometry of Islamic Design (TedEd): http://ed.ted.com/lessons/the-complex-geometry-of-islamic-design-eric-broug

Measuring with Triangles (Scootle): http://www.scootle.edu.au/ec/viewing/L2326/index.html

Just Another Day at the Office (Volvo Ocean Race): https://www.youtube.com/watch?v=aN4CBbeSbP4&feature=youtu.be

World’s Largest Urban Zipline: https://www.youtube.com/watch?v=YcwrRA2BIlw&feature=youtu.be

Similar Triangles and Dynamic Software: https://www.tes.com/teaching-resource/properties-of-similar-figures-and-dynamic-software-11305724

Similar Triangles and introduction to Trigonometry worksheet: https://www.tes.com/teaching-resource/similar-triangles-and-introduction-to-trigonometry-11304321

Labelling sides and setting up SOHCAHTOA (Scootle): http://www.scootle.edu.au/ec/viewing/L6561/asset1.html

Exploring Trigonometric Ratios (Learn Alberta): http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/object_interactives/trigonometry/use_it.html

Trigonometry (Splash ABC): http://splash.abc.net.au/home#!/topic/496568/trigonometry/interactive

Angles of Elevation and Depression Online lesson (MyMaths.co.uk): Angles of Elevation and Depression: http://www.mymaths.co.uk/samples/sampleLessonElevation.swf

Further trigonometry information and interactive practice (Australian Mathematical Sciences Institute (AMSI)): http://www.amsi.org.au/ESA_middle_years/Year9/Year9_md/Year9_2c.html#intro

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 29 of 31

Resources overview

Angles of Elevation and Depression: in the field and in the classroom activity (https://www.tes.com/teaching-resource/angles-of-elevation-and-depression-in-the-field-and-in-the-classroom-11305036)

Area of a Circle, How to get the Formula (mathematicsonline): https://youtu.be/YokKp3pwVFc

Area problem solving samples: https://www.tes.com/teaching-resource/area-of-rectangles-connected-to-triangles-and-special-quadrilaterals-11307354

Area (BBC Bitesize): http://www.bbc.co.uk/bitesize/ks3/maths/measures/area/activity/

Compound shapes (via Scootle): http://www.scootle.edu.au/ec/viewing/L153/index.html

Annulus (Maths is Fun): https://www.mathsisfun.com/geometry/annulus.html

Sector applet (Maths Open Reference): http://www.mathopenref.com/arcsectorarea.html

Shapes glorious shapes’ (ABC Splash): http://splash.abc.net.au/home#!/media/1566372/shapes-glorious-shapes

Renovate, Calculate! (Scootle. DET NSW): http://www.resources.det.nsw.edu.au/Resource/Access/2bb8fd0c-1f62-4658-be9a-6fdae3817fd3/1

Video – How Benjamin Button got his Face (TED Talk): http://ed.ted.com/lessons/how-benjamin-button-got-his-face-ed-ulbrich#watch

‘foil prank’ images for inspiration (https://www.google.com.au/search?q=foil+prank&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjUl8uB1snNAhUEupQKHUa-D4oQsAQIHA&biw=940&bih=475)

variations of cuboid shown here (BBC Bitesize): http://www.bbc.co.uk/bitesize/ks3/maths/shape_space/3d_shapes/revision/3/

printable nets on Scootle: https://www.tes.com/teaching-resource/assorted-nets-6263628

more printable nets on mathworksheets4kids.com: http://www.mathworksheets4kids.com/solid-shapes.php

3D shapes test (BBC Bitesize): http://www.bbc.co.uk/bitesize/quiz/q44861627)

Surface Area using Nets (Khan Academy): https://www.khanacademy.org/math/basic-geo/basic-geo-volume-surface-area/basic-geo-surface-area/e/surface-area

Surface area worksheet (TES): https://www.tes.com/teaching-resource/area-and-surface-area-of-prisms-11315337

Surface Area quiz (Thatquiz.org): https://www.thatquiz.org/tq-4/?-j4080-l5-p0)

Surface area worksheets (Math-aids.com): http://www.math-aids.com/Geometry/Volume/

Design a dome (TryEngineering.com): http://www.tryengineering.org/lessons/designadome.pdf

Cardboard Playhouse (Design Squad Global): https://youtu.be/O8AMJkqo2lw

STEM Resources & Stimulus URLs of linked resources:

Physical World’ - NSW Science K–10: http://syllabus.bostes.nsw.edu.au/science/science-k10/content/982/

The Human Hand (Healthline): http://www.healthline.com/human-body-maps/hand

Teeth (Scootle): http://www.scootle.edu.au/ec/viewing/L730/index.html

Hearing (Scootle): http://www.scootle.edu.au/ec/viewing/L721/index.html

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 30 of 31

Resources overview

Education Resource for Shaun Tan’s The Lost Thing: From Book to Film (ACMI): https://2015.acmi.net.au/media/428194/shaun-tan-ed-kit.pdf

Construction of a 797 Dump Haul Truck: (Education Services Australia): http://www.oresomeresources.com/media_centre_view/resource/video_797_dump_haul_truck_construction/category/mining_videos/section/media/parent/

‘Rectangular Stadium’ (Splash ABC): http://splash.abc.net.au/home#!/media/1454941/

Video – high power line workers.: https://youtu.be/r_1T2_l43Xo

Video – Demolishing the Old Bay Bridge East Span: https://www.youtube.com/watch?v=9xhcTDLM4Ss

Building with Pasta (NASA): https://www.nasa.gov/offices/education/programs/national/summer/education_resources/engineering_grades7-9/E_spaghetti-anyone.html#.V2iVwnozcvk

Navigation Expert Advice (REI): https://www.rei.com/video/2010/10/28/navigation-expert-advicel-triangulation.html

Mapping Farmland using Area and Trigonometry (Splash ABC): http://splash.abc.net.au/home#!/media/86350/mapping-farmland-using-area-and-trigonometry

Landing a plane video (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

Ladders - What are the Rules and Regulations?: http://www.ohsrep.org.au/faqs/ohs-reps-@-work-other-/ladders-what-are-the-rules-and-regulations

‘How Off-roading Works’ (How Stuff Works): http://adventure.howstuffworks.com/outdoor-activities/off-roading/off-roading1.htm

Christo and Jeanne-Claude: http://christojeanneclaude.net/artworks/realized-projects

Carrier Pitching Flight Deck - Landing and Take offs with Bad weather Condition: https://www.youtube.com/watch?v=MI7cYywK-fg

‘Why gold foil is valuable to space exploration’: http://www.geek.com/science/geek-answers-why-does-nasa-use-so-much-gold-foil-1568610/

Food Packaging - Roles, Materials and Environmental Issues: http://www.ift.org/knowledge-center/read-ift-publications/science-reports/scientific-status-summaries/food-packaging.aspx

the use of foil in salons: https://bellatory.com/hair/Hairdressing-How-To-Putting-Foils-in-Hair---Tips--Tricks--Advice-and-Know-How-for-Colouring-Hair-with-Foils

vehicle wrapping: http://solutions.3m.com/wps/portal/3M/en_EU/3MGraphics/GraphicSolutions/Applications/CarWrapping/

Product Safety Australia: https://www.productsafety.gov.au/content/index.phtml/itemId/974027

CHEMM (Chemical Hazards Emergency Medical Management) - Burn Triage and Treatment: https://chemm.nlm.nih.gov/burns.htm

1998 Sydney Hobart Yacht Race film part 1: https://youtu.be/wgsp_kHicu8

Sites showing careers that use maths:

Plus Magazine - career interviews: https://plus.maths.org/content/Career

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 2 Build Make Create Page 31 of 31

Resources overview

Get the Math: http://www.thirteen.org/get-the-math/

Teacher Evaluation of Unit