© NSW Education Standards Authority – Sample unit Mathematics Stage 5 – STEM Advanced Pathway: Formulae Page 1 of 20

Stage 5: Mathematics STEM Advanced Pathway Formulae sample program

Overview Duration

In Formulae students will recall, consolidate and develop the following essential skills:

rearrange formulae

substitute number for variables and solve equations

work with and understand the impact of variable.

In Formulae students will develop the following essential STEM understandings:

that formulae are reliable all-purposes tools.

understanding how variables interact within a formula deepens student understanding of the relationships between different characteristics of the physical world.

spreadsheets are an efficient mathematical tool.

5 weeks

Outcomes

A student:

solves financial problems involving earning, spending and investing money (MA5.1-4NA)

solves financial problems involving compound interest (MA5.2-4NA)

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

applies formulas to calculate the volumes of composite solids composed of right prisms and cylinders (MA5.2-12MG)

applies formulas to find the volumes of right pyramids, right cones, spheres and related composite solids (MA5.3-14MG)

Link to calculus-based course Language/Literacy

Working with formulae that describe the real world enhances student readiness for complex algebraic manipulations in Stage 6. Understanding that formulae describe the relationships between variables enhances student readiness to engage with Calculus in Stage 6. Derivation of one formula from another assists student preparation for proofs and identities in Stage 6.

Key dimensions named: length, width, height, breadth, depth, radius. Distinguish between: area, total surface area, volume and capacity. Units of measurement in one, two and three dimensions. Distinguish between: simple and compound interest, amount, interest, principal and rate.

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Common Misconceptions Pre-test

A student may:

not have developed a sequential method of working through extended problems, as is required for total surface area applications

become confused by the different names for dimensions being interchangeable, for example: height, breadth, width, depth

be able to substitute and solve when the unknown is the subject of the formula, but not when rearrangement is required

be anxious about ability to memorise formulae

not be confident with conversion of units

have trouble interpreting 2D images as 3D objects and hence identifying which values to use in formulae

be challenged by the literacy skills needed to convert worded problems into diagrams

be challenged by the literacy skills needed to identify key information in worded problems

be entering values into calculators incorrectly, for example: fractions and pi

Link to pretest (TES): https://www.tes.com/teaching-resource/prestest-for-area-and-volume-learning-11395929

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Content Teaching and Learning STEM Resources and Stimulus

measure and compare areas using uniform informal units

measure and compare the capacities of pairs of objects using uniform informal units (ACMMG019)

Activity: Set the area activity for one half of the class and the volume activity for the other half.

1. Area one square at a time. Students find the area of a shape by counting the number of square cm it covers on a grid – preferably using a printed image of a scalene triangle. Students could use the ‘Area one square at a time’ digital version (Khan Academy): https://www.khanacademy.org/math/basic-geo/basic-geo-area-and-perimeter/basic-geo-unit-squares-area/e/measuring-area-with-unit-squares ).

2. Volume one cube at a time. Students find the volume of a prism by counting cm cubes required to fill it perfectly – ideally using concrete resources. Students could use the ‘Volume one square at a time’ digital version (Illuminations): http://illuminations.nctm.org/activity.aspx?id=4095. This activity requires students to add single cubes to count volume required to fill shape if the net is folded to form a box).

Link to learning:

Formulae are like ‘life hacks’ for Maths. They save time and reduce counting mistakes. They always work and we don’t need to be able to see an entire object or have a scale diagram in order to be able to calculate answers.

An example image of a Life hack can be found at: ‘Level up’: https://twistedsifter.files.wordpress.com/2013/10/how-to-fit-two-bowls-into-microwave-life-hack.jpg?w=800&h=600

In small groups, students brainstorm all the formulae they can think of – not limited to formulae from school.

Note to teachers – opportunity for informal assessment by

STEM: Being able to ‘choose and use’ a formula correctly is a skill students will need in their futures.

Students consider the wide range of people who use formulae every day.

Some examples:

Science:

Nursing Mathematics – Medical dosage calculators (Nursing Maths): http://www.rncalc.com/

Technology: students read this interview with a fashion designer (Plus Magazine): https://plus.maths.org/content/career-interview-fashion-designer to see the mathematics involved when a thread in one dimension becomes a garment in three dimensions

Engineering:

Article on common automotive mathematical equations – Car Craft Maths (Hot Rod Magazine): http://www.hotrod.com/articles/ccrp-0312-car-craft-math/

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Content Teaching and Learning STEM Resources and Stimulus

observation: student retention of past learning fluency and confidence with mathematical vocabulary students that do and don’t use diagrams without

prompting communication and sharing skills for small group

work

Teachers inform students that while this topic is going to focus on calculating areas, volumes and some financial mathematics, its real objective is to strengthen the students’ ability to observe a situation, decide whether any formulae will be useful and then apply those formulae correctly.

establish and apply for two similar

figures with similarity ratio 1: 𝑘 the

following: matching angles have the same size matching intervals are in the ratio

1: 𝑘 matching areas are in the ratio

1: 𝑘2 matching volumes are in the ratio

1: 𝑘3

solve problems involving similarity ratios and areas and volumes (Problem Solving)

Activity: ‘Exploring space through math’ (NASA): https://www.nasa.gov/audience/foreducators/exploringmath/geometry/Prob_OhChute_detail.html

Note to teachers: ‘Exploring space through math’ provides a lesson plan with student and teacher editions of materials. The Teacher edition includes worked solutions. (The lesson is designed to take 65 minutes and does not include the similarity lesson described below.)

o Table 1 in the lesson plan requires application of scale factor and conversion of linear units.

o Before students attempt Qs 5 –8 they need an opportunity to explore the areas and volumes of similar figures and hence establish those similarity ratios.

A differentiated discovery lesson on similar shapes: ‘Similar shapes inc. area and volume’ can be found at (TES): https://www.tes.com/teaching-resource/similar-shapes-inc-area-and-volume-11017643

STEM: View video: ‘Orion parachutes feature’: https://www.youtube.com/watch?v=eunS46Yfb2Q&feature=youtu.be

Discussion: Can students imagine being astronauts preparing to return to Earth? Can they imagine the number of people and the amount of mathematics it would take for a safe trip through the atmosphere?

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Content Teaching and Learning STEM Resources and Stimulus

Differentiation note: This program combines content from all of the Stage 5 Area and Volume Outcomes. As it is recommended for the commencement of Year 9, teachers should ensure that they select Guided Practice questions appropriate to student readiness. The emphasis is on selecting and applying appropriate formula to answer a question.

Students should practise with questions where the required values for substitution into formulae are apparent. Questions that require additional calculations in order to find values for substitution from other information should be avoided unless students are demonstrating readiness for extension. Questions should be presented in worded and diagram form and only include mixed units once students have developed fluency substituting, solving and rounding with common units.

“Apply Pythagoras' theorem as needed to calculate the volumes of pyramids and cones (Problem Solving) (MA5.3-14MG)” is listed as ‘extension’ Content, as visualising lines within 3D shapes and applying an extended sequence of calculations in order to solve a problem should be reserved for ‘ready’ students.

solve problems involving volume for a range of prisms, cylinders and composite solids (ACMMG242)

find the volumes of solids that have uniform cross-sections that are sectors, including semicircles and quadrants

find the volumes of composite right prisms with cross-sections that may be dissected into triangles and special quadrilaterals

Activity: Design a capsule to live in for 24 hours :Part 1

(This activity could be used as an assessment task – all three parts could be submitted for evaluation and feedback at their conclusion.)

Consider a sleeping pod first – high-altitude mountain climbers risk life-threatening altitude sickness, the only treatment is to get to a lower altitude. Lower altitude can be artificially created by a Gamow Bag.

Students use the digital worksheet: ‘Sleeping pod’ (TES): https://www.tes.com/teaching-resource/survival-pod-part-1-the-sleeping-pod-11394186

Link to learning:

Recognition and naming of prisms and consideration of the properties of prisms – for example – is the bivy bag linked to in worksheet ‘Sleeping pod’ a prism?

Considering an object in space provides opportunity to discuss the interchangeability of the words ‘length’, ‘breadth’,’depth’, ‘height, ‘width’.

STEM: Living in space

Show the first 2:10 of the video: ‘Inside the International Space Station’ https://www.youtube.com/watch?v=bhGydridbEA . This is a 30-minute video. Some students may wish to view the rest of the video in their own time.

Science: In order to design a capsule to live in for 24 hours students must consider two crucial sets of conditions:

1. What does a human body need to survive for 24 hours?

2. What shapes can be combined to allow all 24-hour needs to be met in the smallest total volume?

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Content Teaching and Learning STEM Resources and Stimulus

Consolidation for skill development:

The teacher explicitly teaches:

the names, diagrams and dimensions required for volume calculations for: o solids that have uniform cross-sections that are

sectors, including semicircles and quadrants o right prisms with cross-sections that may be

dissected into triangles and special quadrilaterals

Note to teachers: ideally students had opportunity in Stage 4 Volume topics to understand volume as multiple layers of the base area stacked to form a prism. If not, they should be allowed to experience this with concrete materials if possible. Alternatively a video demonstration could be shown such as: ‘Using layers to find Volume of Rectangular Prisms’: https://www.youtube.com/watch?v=xXoEMkVcWts&feature=youtu.be

Guided practice: School-based and online worksheets could be used as resources.

solve problems involving the volumes of right pyramids, right cones, spheres and related composite solids (ACMMG271)

use the formula to find the volumes of spheres:

– volume of sphere =4

3𝜋𝑟3

– where 𝑟 is the length of the radius

Activity: Design a capsule to live in for 24 hours: Part 2

Students consider themselves in zero gravity. If they were to rotate themselves in every possible direction in a ‘standing’ position, what shaped space will they have traced out?

Would a sphere make a good living space? Students use the worksheet ‘Living Pod” (TES): https://www.tes.com/teaching-resource/survival-pod-part-2-the-living-pod-11394997

Consolidation for skill development:

STEM: recognising and describing the links between existing ideas and observations so that new inventions can be developed from existing knowledge.

Engineering: Show video, ‘Volume of a Sphere’: https://www.youtube.com/watch?v=h4j8l3p22e8&feature=youtu.be Discuss – Compare the efficiency of a spherical and cylindrical living space in zero gravity.

Mathematical Extension:

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Content Teaching and Learning STEM Resources and Stimulus

The teacher explicitly teaches:

the names, diagrams and dimensions required for volume calculations for spheres, and hemispheres.

Guided practice: School-based and online worksheets could be used as resources.

Challenge students to make the connection between the volume of a sphere formula and the sphere compared to a cylinder experiment demonstrated in the video above. (If resources allow, students can conduct this experiment for themselves.)

recognise and use the fact that a pyramid/cone has one-third the volume of a prism/cylinder with the same base and the same perpendicular height, hence, develop and use the formula to find the volumes of right pyramids and right cones:

– volume of pyramid =1

3𝐴ℎ

– volume of a cone is given by

𝑉 =1

3𝜋𝑟2ℎ

– where 𝐴 is the base area, or 𝑟 is the base radius and ℎ is the perpendicular height (Reasoning)

Activity: Discovery of pyramid and cone formulae

(Note to teachers: there are many discovery lessons that allow students to either derive or prove these formulae. If resources do not allow for students to perform the experiment themselves, they could view a video of an example.)

students construct paper (or plastic) cones and the cylinders with equivalent base and height dimensions (or use purchased cones and cylinders)

use water, sand or rice to compare volumes

derive or verify the volume of pyramid and cone formulae

watch the video: ‘Demonstration comparing the formulae for volumes of cylinders and cones’: https://www.youtube.com/watch?v=xwPiA0COi8k&feature=youtu.be

repeat the above steps for a range of pyramids and the prisms with equivalent base and height dimensions

Link to learning:

By exploring the link between cone and cylinder volumes and then seeing that the same relationship exists between pyramid and prism volumes, students can develop their understanding that there are not as many different formulae in Mathematics as they might have thought. The more students understand that formulae can often be derived from one

STEM: Students engage in the scientific process of investigating volume by experimenting with 3D models.

To enhance students’ scientific process, they should not be allowed to accept one result as a proof.

Discuss: How many tests are required to be certain that the relationship they have discovered between cones and cylinders can be applied with reliability?

Note: In Mathematics repeated observances of an empirical result does not constitute proof. Students will be able to prove many of the volume formula when they study the calculus courses in Stage 6 Mathematics.

Mathematics aids Science by providing the statistical support that sufficient experimental evidence has been gathered. Mathematics can then also give a value to how reliable the experimental evidence is.

Once a Mathematical formula has been

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Content Teaching and Learning STEM Resources and Stimulus

another, the more efficient they can become when learning formulae.

Consolidation for skill development:

The teacher explicitly teaches:

names, diagrams and dimensions required for volume of pyramid and cone.

Guided practice:

School-based and online worksheets could be used as resources.

established and proven, it removes the need to keep repeating experiments to get reliable answers. Instead, answers can be calculated.

solve a variety of practical problems related to the volumes and capacities of composite right prisms, cylinders and related composite solids eg: various everyday containers, such as water tanks or cartons used by

removalists (Problem Solving)

find the volumes of composite solids that include right pyramids, right cones and hemispheres, eg find the volume of a cylinder with a cone on top

Activity: Design a capsule to live in for 24 hours: Part 3

Consider you need your survival capsule to fit four people.

Would combining shapes be useful? Students use the digital worksheet: ‘Combining Pods’ (TES): https://www.tes.com/teaching-resource/design-a-survival-pod-part-3-combining-shapes-11395008

Follow-up: students visit NASA’s Space Station site: https://www.nasa.gov/mission_pages/station/main/index.html and in pairs, identify and sketch the 3D shapes used in the construction of: o the Cygnus spacecraft o the Bigelow Expandable Activity Module Extension: Students investigate the Space Station, beginning with the image: https://www.nasa.gov/sites/default/files/thumbnails/image/iss_10-14-16.jpg

Link to learning:

Students have practised identifying simple 3D solids within

STEM: Living in extreme conditions on Earth and using examples of existing habitats to inform design of the survival capsule.

Science: Students ask their Science and HSIE teachers about the extreme conditions that people already live in and what kind of shelter those people require.

Technology: Students find out about Antarctic research stations. What kinds of shelters are used for living and working? Alternatively, students can research this for themselves at the Australian Antarctic Division’s website: http://www.antarctica.gov.au/living-and-working/field-operations/tents-shelters-and-huts

Engineering: Students look at modern, engineered living pods for extreme

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Content Teaching and Learning STEM Resources and Stimulus

more complex, real structures. This prepares them to break diagrams of composite solids into their individual parts for calculation of volume.

Consolidation for skill development:

The teacher explicitly teaches:

examples where one volume is subtracted from another compared to examples when volumes are added.

Guided practice:

School-based and online worksheets could be used as resources.

conditions and see whether they can recognise the shapes of traditional shelters in these.

Discuss with HSIE and/or Science teachers – Who were the first professional ‘Engineers’?

solve a variety of practical problems relating to the volumes and capacities of right pyramids, right cones and spheres

Introduction: Class discussion:

Do students remember the difference between capacity and volume?

Can students explain the difference between capacity and volume using a diagram? Volunteer students can present their explanation with diagram to the class.

Can students think of any examples where the capacity of and volume in a container are very regularly different to one another?

Development:

The teacher:

revises with students conversions between metric units of volume (Stage 4: MA4-14MG)

discusses with the class – As students know the formulae for volume of right pyramids, right cones and spheres, do they also know how to calculate capacity of those 3D

STEM: Share this Cool Australia article ‘Bottled water’ with students: https://www.coolaustralia.org/bottled-water-secondary/

STEM: How does water behave in space?

1. Students ‘think, pair, share’ their opinion.

2. Teacher prompt – What is a meniscus? Why does it form?

3. Ask students whether this prompts them to reconsider or confirm their opinion about how water behaves in space.

4. Show video: ‘Winging out a water-soaked washcloth in space’ (SCA Science): https://www.youtube.com/watch?v=

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Content Teaching and Learning STEM Resources and Stimulus

shapes?

Guided practice:

School-based and online worksheets could be used as resources.

Extension: A challenging question about a conical bottle can be found here (NRICH): http://nrich.maths.org/1945

KFPvdNbftOY&feature=youtu.be

find the dimensions of a particular solid, given its volume, by substitution into a formula to generate an equation, eg find the length of the radius of a sphere, given its volume (Problem Solving)

Note to teachers: the emphasis of this section is the development of student capacity to rearrange and solve an equation, hence, they should not be encouraged to learn ‘new’ formulae for finding dimensions given volume.

The teacher explicitly teaches:

rearranging formulae before and after substitution

Students vote by show of hands which method they think they’ll use, or whether they think there’s no real difference between them.

Activity: Can I fit in here? (Note to teachers - this activity is suitable for homework if the demonstration has been completed.)

Students complete the activity sheet ‘Can I fit in here?’ (TES): https://www.tes.com/teaching-resource/can-i-fit-in-here-11397975

Students view the video: ‘Cats and dogs squeezing into things compilation’: https://youtu.be/4WuGezzNM4w o They are then given the volumes of a rectangular

prism, a cylinder and a hemisphere and must calculate the missing dimension to decide whether they could stand, sit or lie down inside each space.

STEM: (for fun – and thought)

Show students the video.’ Indiana Jones in real life’: https://www.youtube.com/watch?v=qPKKtvkVAjY&feature=youtu.be

Follow with the video: ‘Behind the scenes’: https://www.youtube.com/watch?v=eRZ3izOgqEs&feature=youtu.be (to show no-one was hurt and that it was a planned event). Discuss – What might have gone wrong?

STEM: Thinking – class to give opinions by a show of hands:

Would squeezing into a small space be easier, harder or no different in zero gravity?

Ask volunteers to justify their answers.

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Content Teaching and Learning STEM Resources and Stimulus

Guided practice:

School-based and online worksheets could be used as resources.

apply Pythagoras' theorem as needed to calculate the volumes of pyramids and cones (Problem Solving)

The teacher explicitly teaches examples requiring the use of Pythagoras’ theorem in order to find lengths required by volume formulae for cone and pyramid.

Guided practice: School-based and online worksheets could be used as resources.

use a spreadsheet to graph the value of an investment of a particular amount at various compound interest rates over time (Problem Solving)

apply the simple interest formula to solve problems related to investing money at simple interest rates

find the total value of a simple interest investment after a given time period (Problem Solving)

calculate and compare investments for different compounding periods (Problem Solving)

connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies (ACMNA229)

calculate simple interest using the formula 𝐼 = 𝑃𝑅𝑁 where 𝐼 is the

interest, 𝑃 is the principal, 𝑅 is the interest rate per time period

Activity: Buying a ticket for a space flight – financial planning for a future adventure.

Students visit the XCOR aerospace Space Expeditions site: http://spaceexpeditions.xcor.com/\ to discover that tickets to space can be booked now. (The FAQs page provides a significant amount of mathematical information for discussion.)

With a price tag of $150,000 USD what kind of savings plan would students have to commence in order to afford a ticket for their 30th birthday?

(Note to teachers: this might seem an impossible amount, and yet, students will be faced with greater investment challenges than this if they wish to purchase a home in the future. Realising the power of a compounding investment allowed to run for a long time may be a useful life skill.)

Students view and use the spreadsheet ‘Saving for a ticket to space’ to find the answer (TES): https://www.tes.com/teaching-resource/saving-for-ticket-to-space-spreadsheet-simple-and-compound-interest-activity-11437455

STEM: Space flight as a reality for current students. Already commercial entities are competing for the first tourists in space.

What do students know about this?

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Content Teaching and Learning STEM Resources and Stimulus

(expressed as a fraction or decimal) and 𝑁 is the number of time periods

establish and use the formula to find compound interest: 𝐴 = (1 + 𝑅)𝑛 where 𝐴 is the total

amount, 𝑃 is the principal, 𝑅 is the interest rate per compounding

period as a decimal, and 𝑛 is the

number of compounding periods

o There are two spreadsheets - the first considers simple interest only, the second compares simple and compound interest investments.

Starting with simple interest only, students are instructed to explore the impact of changes to interest rate, initial investment and length of investment individually, and then try combinations of these to get the best result.

Then, students go to the second spreadsheet to see their investment in a simple and compounding account side-by-side.

The investments are automatically graphed to enhance the comparison.

(Note to teachers: calculations focus on the figure $150,000 – not conversion between US$ and AUS$, as this conversion rate is variable and that ‘complication’ is not helpful at this stage of the learning. However, it might make for interesting discussion for some students.)

Extension: As presented, the investment grows from an initial deposit without any further contributions. Students could create a third spreadsheet to track an investment where a regular deposit was added to the account.

Link to learning:

This activity allows students to experience the effectiveness of a spreadsheet for trialling different values in an extensive calculation. It also allows them to see the impact of changes to each variable in an equation on the final result. By presenting simple and compound interest results side-by-side from the beginning, students can realise that the first question in any calculation must be to identify what type of account is involved.

Consolidation for skill development:

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Content Teaching and Learning STEM Resources and Stimulus

Analyse the cell formulae used to calculate simple interest (SI) in the spreadsheet activity: o What is occurring in each cell?

Present formula 𝐼 = 𝑃𝑅𝑁

o Guide students to rearrange the formula to make 𝑃, 𝑅 and 𝑁 the subject. Discuss the usefulness of each arrangement.

Analyse the cell formulae used to calculate compound interest (CI) in the spreadsheet activity: o What is occurring in each cell?

Present formula 𝐴 = (1 + 𝑅)𝑛 o Note the increased complexity of this formula. o Note that this formula gives a different ‘answer’,

discuss the extra step needed to properly compare a simple and compound interest calculation.

o develop ‘formula’ for calculating 𝐼 from C.I formula and 𝐴 from S.I. formula.

o Note for students – rearrangements of the CI formula will be studied in a later topic

Guided practice:

School-based and online worksheets could be used as resources.

solve problems involving surface area for a range of prisms, cylinders and composite solids (ACMMG242)

compare the surface areas of prisms with the same volume

(Problem Solving, Reasoning)

recognise the curved surface of a cylinder as a rectangle and so

(Note to teachers: this next activity should be commenced in the last 10 minutes of a lesson. Equally, it could be conducted in cooperation with the Science faculty).

Activity: ‘Catching your breath in space – the volume inside your lungs’ (think-pair-share)

o The activity is described in the STEM Resources and Stimulus column

(Note to teachers: the ‘machine’ might be as simple as

STEM: Invention and machines

Students are challenged to ‘invent a machine’ that can measure their lung capacity

diagrams and method of use are shared with the class

the class decides on one or two to make in the next Maths lesson

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Content Teaching and Learning STEM Resources and Stimulus

calculate the area of the curved

surface

develop and use the formula to find the surface areas of closed right cylinders: Surface area of (closed) cylinder

= 2𝜋𝑟2 + 2𝜋𝑟ℎ where 𝑟 is the length of the radius and ℎ is the

perpendicular height

solve a variety of practical problems involving the surface areas of cylinders, eg find the area of the label on a cylindrical can

interpret the given conditions of a problem to determine whether a particular cylinder is closed or open (one end only or both ends) (Problem Solving)

blowing up a balloon.)

Link to learning:

In taking lung-capacity measurements using their machines, students have taken air from their lungs and forced it into another shape for measurement – most likely spherical or cylindrical.

When they learn to calculate the surface area of prisms and cylinders they can compare the area housing the displaced volume in their machines to the total surface area of a human lung. This will be a significant demonstration of different surface areas housing equal volumes.

Consolidation for skill development:

Equipment: felt-tip pens, balloons, masking tape and cardboard sheets

Each student inflates and deflates their balloon several times to reduce the balloon’s resistance

Inflate balloons with one lungful

‘Wrap’ balloons in cardboard forcing them into a cylinder shape – tape closed

Trace around the end of the cylinder to create two circles to fit exactly inside it, then compress the balloon as much as possible to reduce empty spaces within the cylinder.

Trim the cylinder to the size of the balloon.

Discuss – Has the volume of the balloon changed? Has the shape of the balloon changed?

Shade the end circles in one colour and the curved surface in another. Colour the edges of the curved surface to match the circles.

Discuss – How many surfaces does a cylinder have? Are all surfaces shaded right now?

and plans the materials to bring (or teacher arranges for materials)

in a subsequent lesson the ‘machines’ are built and used to measure lung capacity. Students sketch the ‘machine’ before and after the experiment to show how the measurement is conducted.

Science: (May be completed in cooperation with the Science faculty.)

The total surface area of human lungs is a surprise to most students. Have students look this up, and then try to think of a familiar thing of approximately the same area to aid their visualisation.

STEM – Space flight: Students investigate breathable air on the International Space Station.

How much air is there?

How it is stored?

Is it recycled?

Can they ‘make’ air in space?

Why can’t 100% oxygen be used?

What are the components of air?

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Content Teaching and Learning STEM Resources and Stimulus

Take cylinder apart and trim away any unshaded part of the curved surface.

Identify the two circles and one rectangle and develop the formula

Surface area of (closed) cylinder = 2𝜋𝑟2 + 2𝜋𝑟ℎ

Calculate the total surface area and compare to the total surface area of human lungs.

Discuss – if one or both ends of the cylinder were removed, would the volume change? o What about the total surface area?

Develop the formula for total surface area of a cylinder with one and two open ends.

Guided practice:

School-based and online worksheets could be used as resources.

Differentiated questions at four difficulty levels can be completed here: ‘Surface area of a cylinder’ (Prodigi): https://campus.mangahigh.com/en-au/px/396/0/0

visualise and sketch the nets of right prisms

interpret the given conditions of a problem to determine the number of surfaces required in the calculation (Problem Solving)

apply properties of geometrical shapes to assist in finding areas, eg symmetry

Activity: Flat pack

The furniture industry has been changed by ‘flat pack’ companies. Ikea would be the most well-known.

The net of a 3D object is essentially its ‘flat-pack’.

Equipment: recycled packaging such as Toblerone boxes and tea boxes and coloured felt-tip pens

Students use a thick marker to mark each edge of the boxes in different colours

Sketch each box showing the colours of the edges visible in their sketches

STEM: Show the video ‘Star Trek Beyond: Inside the amazing visual effects of the Starship Enterprise’ (Wired UK): https://www.youtube.com/watch?v=LgZLggOGZxo&feature=youtu.be

Compare to a clip from the original such as the video ‘Accidental injection': https://www.youtube.com/watch?v=a5kICRWh3n8&feature=youtu.be&list=PLufIO1FTWFz9esRDQVIUfdyF0lR5zCrp0

Discuss – How have people in STEM careers changed the way we see space

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Content Teaching and Learning STEM Resources and Stimulus

Cut boxes into their nets

Sketch nets showing colours of edges

Calculate the area of each face and hence find the total surface area of each box

Discussion – Did every individual face have to be calculated or were there short cuts for some of the boxes?

Link to learning:

Identifying and matching edges on a prism and its net connects the 3D solid to the 2D version of its area. Students are graduated from this to being able to identify important dimensions of faces without requiring the net.

Consolidation for skill development:

Provide students with an explicit structure for calculating the surface area of right prisms with an emphasis on naming and working with prisms in different orientations.

A set of formulae with diagrams ‘3D & 2D shapes’ can be found here: http://staff.argyll.epsb.ca/jreed/math8/strand3/3208.htm. This demonstrates using ‘base perimeter × height’ for combined area of prism ‘sides’. (Note: the included pyramids are not required for this course.)

Guided practice:

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

Complete the worksheet: ‘Surface Area and Volume of Prisms’ (TES): https://www.tes.com/teaching-resource/surface-area-and-volume-of-prisms-6281186

Past NAPLAN Numeracy questions involving nets and 3D

in film and television? How do these presentations compare to the current reality.

Think – Is there any furniture on the International Space Station? Are there plates, cups and cutlery?

Students discuss their opinion and then research for an answer.

© NSW Education Standards Authority – Sample unit Mathematics Stage 5 – STEM Advanced Pathway: Formulae Page 17 of 20

Content Teaching and Learning STEM Resources and Stimulus

figures

find the surface areas of composite solids involving right prisms and cylinders

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

calculate the area of an annulus (Problem Solving)

solve a variety of practical problems related to surface areas of prisms, cylinders and related composite solids, eg compare the amount of packaging material needed for different objects

Consolidation for skill development and guided practice:

The teacher provides the students with an explicit structure for calculating the surface area of composite prisms and cylinders with an emphasis on identifying when faces are added or subtracted (covered up).

Guided practice:

School-based and online worksheets could be used as resources.

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

The teacher explicitly teaches examples requiring the use of Pythagoras’ theorem when the lengths required to calculate the TSA formulae of triangular prisms are not presented in the question.

Guided practice: School-based and online worksheets could be used as resources.

Assessment strategies

Activity: Students could be allowed to select one of the learning activities and submit a report including diagrams and calculations for assessment.

Topic Test: Short answer and multiple choice test.

© NSW Education Standards Authority – Sample unit Mathematics Stage 5 – STEM Advanced Pathway: Formulae Page 18 of 20

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

The following URLs are correct at the time of publication:

Additional Teacher Resources URLs:

Link to pretest (TES): https://www.tes.com/teaching-resource/prestest-for-area-and-volume-learning-11395929

Video: ‘Himalayan Trexplorers: The Gamow Bag’: https://youtu.be/Nw08GKEAht0

Japanese capsule hotels: https://en.wikipedia.org/wiki/Capsule_hotel

Bivy sacks: http://adventurefriends.com.au/bivy-sacks-ferrino-high-quality-portable-tents

A-frame tent images:

https://www.google.com.au/search?q=a+frame+tent&biw=1113&bih=501&tbm=isch&source=lnms&sa=X&ved=0ahUKEwiFyZ7fyODPAhVEvY

8KHSRkD9MQ_AUIBigB

Abu Dhabi airport sleeping pods: http://wonderfulengineering.com/abu-dhabi-airport-installs-worlds-first-gosleep-sleeping-pods/

Tentsile tents: https://youtu.be/Hgx6lN5F6g4

Spherical Treehouses: http://freespiritspheres.com/

Shelter domes as alternative housing: http://pacificdomes.com/shelter-domes/

Apple huts: http://www.antarctica.gov.au/living-and-working/field-operations/tents-shelters-and-huts/larger-shelters

Yurt: https://en.wikipedia.org/wiki/Yurt

Image of a simple pitched roof cottage: http://www.bylerbarns.com/wp-content/uploads/2016/08/8x12-Vinyl-Cottage-Shingles-1-1030x687.jpg

Image of a small castle: https://s-media-cache-ak0.pinimg.com/originals/b9/0e/06/b90e06db8424c26d832ebc980d5fdaa7.jpg

World’s deepest diving glass dome mini submarine: https://www.gps4us.com/news/post/Worlds-deepest-diving-glass-dome-mini-submarine-

20111219.aspx

Teaching & Learning URLs of linked resources:

© NSW Education Standards Authority – Sample unit Mathematics Stage 5 – STEM Advanced Pathway: Formulae Page 19 of 20

Resources overview

‘Area one square at a time’ digital version (Khan Academy): https://www.khanacademy.org/math/basic-geo/basic-geo-area-and-

perimeter/basic-geo-unit-squares-area/e/measuring-area-with-unit-squares

‘Volume one cube at a time’ digital version (Illuminations): http://illuminations.nctm.org/activity.aspx?id=4095

Life hack image: ‘Level up’: https://twistedsifter.files.wordpress.com/2013/10/how-to-fit-two-bowls-into-microwave-life-hack.jpg?w=800&h=600

‘Exploring space through math’ (NASA): http://www.nasa.gov/audience/foreducators/exploringmath/geometry/Prob_OhChute_detail.html

‘Similar shapes inc. area and volume’ (TES): https://www.tes.com/teaching-resource/similar-shapes-inc-area-and-volume-11017643

‘Sleeping Pod’ (TES): https://www.tes.com/teaching-resource/survival-pod-part-1-the-sleeping-pod-11394186

Video: ‘Using layers to find Volume of Rectangular Prisms’: https://www.youtube.com/watch?v=xXoEMkVcWts&feature=youtu.be

‘Living Pod’ (TES): https://www.tes.com/teaching-resource/survival-pod-part-2-the-living-pod-11394997

Video: ‘Demonstration comparing the formulae for volumes of cylinders and cones’:

https://www.youtube.com/watch?v=xwPiA0COi8k&feature=youtu.be

‘Combining Pods’ (TES): https://www.tes.com/teaching-resource/design-a-survival-pod-part-3-combining-shapes-11395008

NASA’s Space Station (NASA): https://www.nasa.gov/mission_pages/station/main/index.html

Image of the Space Station (NASA): https://www.nasa.gov/sites/default/files/thumbnails/image/iss_10-14-16.jpg

‘Conical bottle’ (NRICH): http://nrich.maths.org/1945

‘Can I fit in here?’ (TES): https://www.tes.com/teaching-resource/can-i-fit-in-here-11397975

Video: ‘Cats and dogs squeezing into things compilation’: https://youtu.be/4WuGezzNM4w

XCOR aerospace Space Expeditions: http://spaceexpeditions.xcor.com/\

Spreadsheet ‘Saving for a ticket to space’ (TES): https://www.tes.com/teaching-resource/saving-for-ticket-to-space-spreadsheet-simple-and-

compound-interest-activity-11437455

‘Surface area of a cylinder’ (Prodigi): https://campus.mangahigh.com/en-au/px/396/0/0

A set of formulae with diagrams ‘3D & 2D shapes’: http://staff.argyll.epsb.ca/jreed/math8/strand3/3208.htm

‘Surface Area and Volume of Prisms’ (TES): https://www.tes.com/teaching-resource/surface-area-and-volume-of-prisms-6281186

‘Surface area of composite shapes’: https://www.sophia.org/tutorials/surface-area-of-composite-figures

STEM Resources & Stimulus URLs of linked resources:

Medical dosage calculators (Nursing Maths): http://www.rncalc.com/

Interview with a fashion designer (Plus Magazine): https://plus.maths.org/content/career-interview-fashion-designer

Common automotive mathematical equations – Car Craft Maths (Hot Rod Magazine): http://www.hotrod.com/articles/ccrp-0312-car-craft-

© NSW Education Standards Authority – Sample unit Mathematics Stage 5 – STEM Advanced Pathway: Formulae Page 20 of 20

Resources overview

math/

Video: ‘Orion parachutes feature’: https://www.youtube.com/watch?v=eunS46Yfb2Q&feature=youtu.be

Video: ‘Inside the International Space Station’ https://www.youtube.com/watch?v=bhGydridbEA

Video: ‘Volume of a Sphere’: https://youtu.be/h4j8l3p22e8

Australian Antarctic Division: http://www.antarctica.gov.au/living-and-working/field-operations/tents-shelters-and-huts

‘Bottled water’ (Cool Australia): http://www.coolaustralia.org/bottled-water-secondary/

Video: ‘Wringing out a water-soaked washcloth in space’ (SCA Science): https://www.youtube.com/watch?v=KFPvdNbftOY&feature=youtu.be

Video: ‘Indiana Jones in real life’: https://www.youtube.com/watch?v=qPKKtvkVAjY&feature=youtu.be

Video: ‘Behind the scenes’: https://www.youtube.com/watch?v=eRZ3izOgqEs&feature=youtu.be

Video: ‘Star Trek Beyond: Inside the amazing visual effects of the Starship Enterprise’ (Wired UK): https://youtu.be/LgZLggOGZxo

Video: "Accidental injection': https://youtu.be/a5kICRWh3n8?list=PLufIO1FTWFz9esRDQVIUfdyF0lR5zCrp0

Sites showing careers that use maths:

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

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

Teacher Evaluation of Unit