mathematics units grade 12 foundation - csomathscience · pdf fileuse of graphics calculator...

Mathematics units Grade 12 foundation

Contents

12F.1 Using algebra 211 12F.7 Measures 257

12F.2 Geometry 1 219 12F.8 Random variables and probability 263

12F.3 Statistics 225 12F.9 Functions 3 275

12F.4 Functions 1 235 12F.10 Using vectors 281

12F.5 Geometry 2 243 12F.11 Functions 4 289

12F.6 Functions 2 251

Mathematics scheme of work: Grade 12 foundation units 90 teaching hours

UNIT 12F.1: Using algebraConsolidating work with symbolsRearranging harder formulaeGenerating formulae from physical situationsGenerating recursive sequences to model real-world applications8 hours

UNIT 12F.8: Random variables and probabilityEmpirical probability; theoretical probability models;riskCombined events; addition and multiplication rules;tree diagramsSimulations using random variables or diceUsing ICT12 hours

25% 25%

1st semester45 hours

2nd semester45 hours

UNIT 12F.2: Geometry 1Using ICT to investigate geometryGenerating patterns; congruence andsimilarity; geometric constructions;plans and elevations4 hours

UNIT 12F.5: Geometry 2Transformation of rectilinear figures;translations, rotations, enlargements;combining transformationsMaps and scale drawings; plans andelevationsUse of ICT7 hours

UNIT 12F.7: MeasuresFinding lengths, areas, volumesUsing approximations to calculate areasand cross-sections of irregular shapesSI unitsProblems involving compound measuresand rates4 hours

Reasoning and problem

solving should be integrated into each unit

UNIT 12F.0: Grade 11F revision3 hours

UNIT 12F.4: Functions 1Standard functions: linear, quadratic, cubic;reciprocal function; sine and cosine functionsModulus function and other non-standardfunctionsUse of graphics calculator12 hours

UNIT 12F.6: Functions 2Combinations of functions; composite functionsInverse functions and their graphsDeconstructing functions6 hours

50%

UNIT 12F.10: Using vectorsCoordinate grids, position vectors, unitvectors, componentsAdding two vectors; multiplying a vectorby a scalar; scalar productMagnitude of a vectorAngle between two vectors7 hours

UNIT 12F.3: StatisticsData types, data collection and sampling; primaryand secondary sources; surveys andquestionnairesMeasures of central tendency and spread; movingaveragesRelative frequency histograms; cumulativefrequency distributionsMaking inferences and presenting findings usinggraphs, charts and tables11 hours

UNIT 12F.9: Functions 3Transforming functionsCombinations of functions and their graphs6 hours

UNIT 12F.11: Functions 4Exponential curves; growth and decayExponential and logarithmic functionsUsing calculator function keys to plot graphs, findvalues and solve related equationsModelling real-world situations10 hours

211 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.1 | Using algebra © Education Institute 2005

GRADE 12F: Using algebra

Formulae and sequences

About this unit This unit is the first of five units on algebra for Grade 12 foundation. It offers more work on manipulation and formulae.

The unit is designed to guide your planning and teaching of mathematics lessons. It provides a link between the standards for mathematics and your lesson plans.

The teaching and learning activities should help you to plan the content and pace of lessons. Adapt the ideas to meet your students’ needs. Supplement the activities where necessary with appropriate tasks and exercises from textbooks and other resources, including ICT.

For consolidation activities, look at the units for Grade 11 foundation; for extension or enrichment, consider activities in the Grade 12 advanced units, or on those websites referred to in the text.

Introduce the unit to students by summarising what they will learn and how this builds on earlier work. Review the unit at the end, drawing out the main learning points, links to other work and real-world applications.

Previous learning To meet the expectations of this unit, students should already be able to manipulate simple algebraic expressions and formulae and be familiar with geometric sequences.

Expectations By the end of the unit, students will identify and use connections between mathematical topics. They will develop and explain chains of logical reasoning, using correct mathematical notation and terms. They will make appropriate use of their knowledge of number sets, and their calculation skills, established in Grades 10 and 11. They will rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts. They will generate recursive sequences to model the behaviour of real-world situations.

Students who progress further will gain increased confidence in manipulative skills and have a greater awareness of the algebraic structure that lies behind practical examples. They will multiply, factorise and simplify expressions, divide a polynomial by a linear or quadratic expression and use the remainder theorem. They will understand the concept of a composite function.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access and computer linked to data projector • spreadsheet software such as Microsoft Excel • computers with spreadsheet software and Internet access for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • dividend, divisor, quotient, remainder, factor • series, sigma notation, recursive definition

UNIT 12F.1 8 hours


Standards for the unit

8 hours SUPPORTING STANDARDS

Grade 11F standards CORE STANDARDS Grade 12F standards

EXTENSION STANDARDS Grade 12F and 12AQ standards

12F.1.3 Identify and use interconnections between mathematical topics.

12F.1.6 Develop longer chains of logical reasoning, using correct mathematical notation and terms.

12F.2.1 Make appropriate use of their knowledge of number sets from Grades 10 and 11.

11F.3.1 Understand and use the laws of exponents to calculate and simplify problems, including mental calculations in appropriate cases.

12F.3.1 Develop further confidence in all the calculation skills established in Grades 10 and 11.

12F.5.5 Understand the concept of a composite function and use the notation y = f(g(x)).

11F.4.4 Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions, collecting and simplifying similar terms.

12AQ.2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

2 hours

Series notation and formulae

4 hours

Algebraic manipulation

2 hours

11F.4.5 Factorise expressions of the form a2x2 – b2y2, and quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions.

12AQ.2.3 Understand and use the remainder theorem.

11F.4.6 Simplify numeric and algebraic fraction expressions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

11F.4.7 Generate further formulae from a physical context, and rearrange formulae connecting two or more variables; substitute an expression for a given variable into a different formula containing this variable.

12F.4.1 Rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts.

11F.4.1 Know the properties of geometric sequences and the conditions under which an infinite geometric series can be summed.

12F.4.2 Generate recursive sequences from term-to-term and position-to-term definitions to model the behaviour of real-world situations, for example population growth.

Unit 12F.1


Activities

Objectives Possible teaching activities Notes School resources

2 hours

Series notation and formulae Generate sequences from term-to-term definitions and from position-to-term definitions, including recursive sequences, to model the behaviour of real-world situations, for example population growth.

Series notation

Discussion Begin by explaining sigma notation as a compact way to present the sum of a set of numbers or the terms of a sequence. Do this first by a number of simple examples, and then displayed on a spreadsheet column, where the row function can replace the index on sigma.

Conclude the presentation with a number of examples worked orally, turning series into sigma notation with index and limits, and vice versa.

Examples • Express in full:

6

1( 1)

=

+∑rr r

• Use sigma notation to express 7 6 55 4 3

2.+ + +

6

1( 1) on a spreadsheet

r

r r=

+∑

A1=row(A1)*(row(A1)+1)

B1=2; B2=B1+A2

A1 and B2 are replicated down the columns.

Extend discussion using the applet Sequences (www.fi.uu.nl/wisweb/welcome_en.html).

This column is for schools to note their own resources, e.g. textbooks, worksheets.

Practice Get students to use the row function in a spreadsheet to simulate the index in the sum as shown above. Many textbook exercises designed for pencil and paper can be adapted to this. Extend such exercises to incorporate the formulae for the sum of the first n natural numbers, the sum of their squares, and the sum of their cubes. • Express 5 × 6 + 6 × 7 + 7 × 8 + 8 × 9 + … + 199 × 200 using sigma notation, and hence, with

the aid of a spreadsheet, evaluate the sum. • Use replication to generate the first 50 terms of the series 5, 5, 10, 15, 25, … Find its sum.

• Use replication to show the terms of 100

1 1r

rr= +∑ on a spreadsheet; hence calculate the sum.

• If 2 16

1( 1)(2 1),

n

r

r n n n=

= + +∑ find a formula for 2

2

1.

n

r n

r= +∑

• If un = n(n + 1)(n + 2), write down and simplify an expression for un+1 / un, and hence obtain a

recurrence relation for the sequence (un). • Use a spreadsheet to show the growth of a sum of money invested at compound interest,

e.g. QR 1 000 000 at 4% per annum.

Exercises like these promote new ways of looking at this sort of problem. They can lead to applications where algebra models situations such as accumulating bank deposits, population growth, appreciation and decay.

Use the applet Discrete dynamic models from www.fi.uu.nl/wisweb/welcome_en.html to explore and discuss some real-world situations.

Unit 12F.1



Algebraic manipulation: review and extension Students can make progress on this at various levels. Here are some starting points.

On the web MathsNet has interactive pages on many of these algebraic topics.

4 hours

Algebraic manipulation

Make appropriate use of their knowledge of number sets from Grades 10 and 11.

Develop further confidence in all the calculation skills established in Grades 10 and 11.

Develop longer chains of logical reasoning, using correct mathematical notation and terms.

Identify and use interconnections between mathematical topics.

Expanding quadratics Practise expansions such as (3x + 4)(x – 3). This is covered at various levels in several other grades. It can be approached through games, diagrams, or just routine practice.

Less able students can find this work both frustrating and satisfying, since it requires concentration but also gives precise and convincing answers. It adapts itself well to short class quizzes (practising essential mental and algebraic skills) and to diagrams.

Expansion of brackets is on MathsNet at www.mathsnet.net/asa2/modules/ p12pracbrac.html.

Geometrical illustration of an algebraic product

from Unit 11F.8

Factorisation of trinomials Factorising expressions such as 2x2 – 3x – 2 make good puzzles for brainstorming solutions, since factorisation is a challenge for those who have newly acquired confidence in algebraic manipulation. Students can be tested against software programs that will expand the results as a check. Alternatively, students may work in pairs, providing answers orally to each other. A further possibility is to ask them to make a short quiz to exchange with other students.

This is on MathsNet at www.mathsnet.net/asa2/modules/ p12pracfac.html.

Simplification

Simplification of algebraic fractions such as 2

23 2

2 2x xx− +

− depends on factorisation skills. Extend

the work, beginning from simple fractions with factors to cancel and extending to sums, differences, products and quotients to simplify.

This is harder work, suitable only for those who are already confident in the underlying skills.

Simplification is on MathsNet at www.mathsnet.net/asa2/modules/p21factor.html.

Model solution using factorisation: 2

23 2 ( 2)( 1)

2 2 2( 1)( 1)2

2( 1)

x x x xx x x

xx

− + − −=− − +

−=+

Solving equations Students met the solution of linear equations and quadratics in Grade 11F. Revision of those two topics can be extended to work on more complex problems dependent on simplifying skills and on algebraic fractions.

• (x – 1)(x – 2) – (x – 3)(x – 4) = (x – 1)(x – 4)

• 1 2 11

x xx x− −+ =

−

Quadratics were covered in Unit 11F.10.


Objectives Possible teaching activities Notes School resources Indices

Revise the rules for positive integral indices. Have students discuss how to give meaning to: • x–6 (and other examples with negative integer indices); • 161/4 (and other examples involving fractional indices); • 50.

Include revision of standard form.

Depending on students’ ability and achievement level, allow time to practise and consolidate the rules implicit in these examples, or work on harder problems. Harder examples are:

• Is ( )223 364 (64 ) ?= Does this generalise?

• Simplify (3 3⁄8)–2/3.

• Solve 8x = 644x–2.

Surds Work on surds covers: • expressing roots of integers in terms of the smallest possible surds; • simplifying expressions that involve surds; • arithmetic operations that include the appearance of surds as factors in the denominators.

On the web Practice on surds is on MathsNet at www.mathsnet.net/asa2/modules/ p12ratsurd2.html.

An example of a harder surd problem is:

3 2 2 3 2 2 3 5 23 5 2 3 5 2 3 5 2

29 21 241

+ + += ×− − +

− −=

Pascal’s triangle Pascal’s triangle was the basis of expansions of (1 + x)3, (1 + x)4, etc., in Grade 11F. Extend this now to expansions of (a + b)3, (a + b)4, etc., using a variety of expressions for a and b.

Examples • Expand (1 + 2x)5.

• Find the term independent of x in the expansion of 5

23

12

xx

⎛ ⎞+⎜ ⎟⎝ ⎠

.

• Find the term in x3 in the expansion of (1 + x + 2x2)6.

There are lots of websites containing interactive versions of Pascal’s triangle which help students to find the patterns, e.g. mathforum.org/ workshops/usi/pascal/mo.pascal.html.

Long division and the remainder theorem This is covered in Unit 12AQ.1 of Grade 12AQ. This is extension work on the same skills but with considerable development.

On the web MathsNet’s interactive cubic factorisation page is at www.mathsnet.net/asa2/2004/ c12cubicfactor_2.html.

This is definitely extension work.



Substitution

Discussion Letters and brackets in the algebraic structure of expressions are equivalent. Extend the idea to cases where an expression can be substituted for a letter (so that it is effectively turned into a bracket) before simplification. In effect, one formula can be substituted into another.

Construct exercises to practise the skill.

2 hours

Rearranging formulae

Discussion The discussion of technique can now take a change of direction. The rules of priority are well established; the solution of equations, use of roots and the simplification of expressions have all been practised. These ideas become more abstract when applied to the solution of algebraic equations.

Demonstrate lots of worked examples. As the discussion progresses, expect to make more use of students’ suggestions; these may sometimes lead to indirect approaches that still give the same conclusion, so that more efficient methods can be established.

The best strategy is to: • square up roots; • remove fractions; • multiply out brackets; • collect terms; • factorise; • divide.

Examples

2 2

2 2

2

2

2 ; find .

2 4

44

lT gg

l lT Tg g

gT llg

T

π

π π

ππ

=

⎛ ⎞= ⇒ = ⎜ ⎟

⎝ ⎠⇒ =

⇒ =

( 1) ; find .( 1)

(1 )

1 1

x x y xx x y x xy y

x xy yx y y

y yxy y

= −= − ⇒ = −

⇒ − = −⇒ − = −

−⇒ = =

− −

Discussion and practice

Routine practice of this type of question should follow. The work done will require either thorough correction or thorough discussion, since there are often equivalent answers and any mistakes need careful follow-up.

The examples considered should make as much use as possible of formulae which may be familiar to students from scientific contexts (such as the expression for the time period of a simple pendulum), so that they may see the relevance of such formulae in real-world applications.

Example

Find R in terms of R1 and R2 when 1 2

1 1 1R R R

= + .


Assessment

Examples of assessment tasks and questions Notes School resources

Farida is making a scale model of the Earth and the Moon for a museum. She has found out the diameters of the Earth and the Moon, and the distance between them in metres.

Diameter of the Earth 1.28 × 107 m

Diameter of the Moon 3.48 × 106 m

Distance between Earth and Moon 3.89 × 108 m

a. How many times bigger is the diameter of the Earth than the diameter of the Moon?

b. In Farida’s scale model the diameter of the Earth is 50 cm. What should be the distance between the Earth and the Moon in Farida’s model?

Assessment

Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities.

Look at the table.

Earth Mercury

Mass (kg) 5.98 × 1024 3.59 × 1023

Atmospheric pressure (N/m2) 2 × 10–8

The atmospheric pressure on Earth is 5.05 × 1012 times as great as the atmospheric pressure on Mercury. Calculate the atmospheric pressure on Earth.

What are all values of x for which the inequality 5x + 5⁄3 ≤ –2x – 2⁄3 is true?

A. x ≤ −7⁄9

B. x ≤ −1⁄3

C. x ≥ 0

D. x ≥ 7⁄3

E. x ≥ 9⁄3

TIMSS Grade 12

The value of a new car depreciates by 20 per cent at the end of the first year. It then loses value at the rate of 10 per cent for every subsequent year. Set up a formula to describe the value V of the car t years after purchase. After how many years will the car be worth a quarter of its purchase price?

In a TV quiz show a contestant can triple her winnings if she survives from one round to the next. Write a formula for her prize Pn+1 in the (n + 1)th round in terms of her prize Pn in the nth round.

Write an alternative formula for Pn+1 in terms of n.

The prize for winning in the first round is QR 1000. What is the minimum number of rounds that will have to be contested to win at least QR 700 000?

Unit 12F.1



At the end of every year a car loses 30 per cent of its value at the start of the year. Construct a formula, in terms of the original purchase price, to give the value of the car n years after purchase. After how many years will the car first be worth less than 20 per cent of its original value?

The sum of the infinite geometric series 1 – 1⁄2 + 1⁄4 – 1⁄8 + … is:

A. 5⁄8

B. 2⁄3

C. 3⁄5

D. 3⁄2

E. ∞

TIMSS Grade 12

A sequence is defined by un+2 = un+1 – un with u1 = 5 and u2 = –4.

Write the first eight terms of the sequence.

A sequence is defined by un = n(n – 1) + 41.

Write the first twelve terms of this sequence. What do you notice about these terms?

Form a conjecture about this sequence and carry out further tests to see if your conjecture is correct.

Find r +∑10

1( 1).

Find r∑ 2 for integer values of r from 1 to 10.

Rewrite these sums using sigma notation:

72 + 82 + 92 + 102 1⁄121 – 1⁄144 + 1⁄169 – 1⁄196 + 1⁄225

219 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.2 | Geometry 1 © Education Institute 2005

GRADE 12F: Geometry 1

Patterns and constructions

About this unit This unit is the first of two on geometry for Grade 12 foundation. It extends work on congruence and similarity using a dynamic geometry system (DGS), and work on plans and elevations.





Previous learning To meet the expectations of this unit, students should already be able to use a dynamic geometry system, work with concepts such as transformation, similarity and congruence, and draw plans and elevations.

Expectations By the end of the unit, students will solve routine and non-routine problems. They will use a range of strategies to solve problems, including breaking down complex problems into smaller tasks. They will aim to generalise. They will synthesise, present, interpret and criticise mathematical information. They will draw and use plans and elevations. They will explore aspects of geometry using ICT.

Students who progress further will connect their work with the challenge of proof.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access and computer linked to data projector • dynamic geometry system (DGS) such as:

Geometer’s Sketchpad (see www.keypress.com/sketchpad) Cabri Geometrie (see www.chartwellyorke.com/cabri.html)

• computers with dynamic geometry system software for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • congruence, similarity • projection, plan, elevation • perpendicular bisector, angle bisector, altitude, median, circumcentre,

orthocentre, centroid, in-centre

UNIT 12F.2 4 hours




Grade 9 and 11F standards CORE STANDARDS Grade 12F standards

EXTENSION STANDARDS Grade 12A standards

12F.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

12F.1.4 Break down complex problems into smaller tasks.

12F.1.5 Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

12F.1.9 Generalise when appropriate.

2 hours

Investigating properties using DGS

2 hours

Problem solving with plans and elevations

12F.1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

11F.6.1 Use dynamic geometry systems to conjecture results and to explore geometric proof.

12F.6.1 Use ICT to investigate a range of geometrical situations, including: • the generation of geometric patterns, including Islamic patterns; • similarity and congruence; • constructions; • plans and elevations.

9.5.9 Recognise 3-D objects from 2-D representations; draw the plan and elevation of a 3-D object from sketches and models; sketch or build a 3-D object given its plan and elevation.

12F.7.4 Draw the plan and elevation of two-dimensional projections of three-dimensional rectilinear objects.

Unit 12F.2


Activities


Investigation 1: constructions Use a dynamic geometry system (DGS) such as Geometer’s Sketchpad to revise the basic constructions: • a perpendicular bisector of a line segment; • an angle bisector; • a perpendicular from a point to a line.

Ask students how DGS does these things. At the outset make clear that the tools for geometrical development here comprise only: • a straight edge (i.e. no distance measurement on a ruler); • a pair of compasses; • a pencil.

Students may suggest acceptable if non-standard ways of doing these constructions; use their contributed ideas to draw out all the standard procedures, either as explicit suggestions or as equivalent processes.

On the web There is an online course in geometrical construction at MathsNet (www.mathsnet.net/campus/construction).

At illuminations.nctm.org/tools/index.aspx there is a useful applet Interactive geometry dictionary: Lines in geometry.


2 hours

Investigating properties using DGS

Use ICT to investigate a range of geometrical situations, including: • the generation of

geometric patterns, including Islamic patterns;

• similarity and congruence; • constructions; • plans and elevations.

Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

Generalise when appropriate.

Give students time to practise their construction skills in DGS. Give them simple examples of the three construction procedures, together with more complex processes.

An example of a more complex process is: • draw a triangle; • construct the perpendicular bisectors of its three sides; • discover the bisectors’ concurrency at the circumcentre.

There are similar cases of: • the orthocentre where the three altitudes meet; • the centroid where the three medians meet; • the in-centre where the three angle bisectors meet.

Extension for able students This practical exercise can be extended to the Euler line, the line on which the circumcentre, the orthocentre and the centroid lie. Proving that result is a considerable challenge.

A circumcircle produced in Sketchpad.

Unit 12F.2



Investigation 2: tilings Use DGS to investigate the scope for tiling a plane using different shapes, for example: • an equilateral triangle; • an arbitrary triangle; • a pentagon; • a hexagon.

This can be extended to combinations of shapes.

There are other interesting tessellations at: • www.cromp.com/tess/home.html; • britton.disted.camosun.bc.ca/jbsymteslk.htm.

On the web

The tessellation of fish above is from www.cromp.com/tess/home.html.

Investigation 3: Islamic patterns Give students a starting point such as the following. • Make a circle with centre at a point on a line. • Make two circles with centres at the points of intersection. • Make four circles with centres at the new points of intersection. • Continue outward in all directions. Discuss the symmetries in the patterns produced and the transformations involved. Other designs are possible.

Follow up by using the applet on www.cgl.uwaterloo.ca/~csk/washington//taprats.

In the applet, students start with a tiling of the plane made up of regular polygons. The polygons are filled with radially symmetric motifs like those found in the Islamic tradition. The tiles forming the gaps between the regular polygons are then filled in by finding natural extensions of the lines meeting their boundaries.

Further reading for students:

Syed Jan Abas and Amer Shaker Salman (1995) Symmetries of Islamic Geometrical Patterns. World Scientific (see www.ethnomath.org/resources/abas2001.pdf).

On the web



Harder drawings Put this problem to students: • Draw the plan, and front and side elevations, of a pyramid VABCD standing on a base ABCD

6 cm square and having slant edges 10.5 cm long, if two edges of the base make an angle of 20° with the plan’s baseline.

Ask students how they will start. The response has to be to begin with the base. This must be drawn so that one side is angled at 20°. Use Sketchpad (or other version of DGS) to draw this, with students providing prompts.

Once the square is drawn, ask students for the next step. The response this time must be to work out the height of the pyramid that is not given directly. Do this by constructing an isosceles triangle with side 10.5 cm on one side of the square.

Show how to begin the elevation constructions by constructing perpendiculars and parallels to the baseline, and (arcs of) circles to transfer the measurements. Then finish the construction.

Get students to do all this in Sketchpad as a small project.

The plan and elevation as produced in Sketchpad.

A dynamic geometry system removes much of the work in such a construction, and makes students think more about what they are doing to complete all the steps.

2 hours

Problem solving with plans and elevations Draw the plan and elevation of two-dimensional projections of three-dimensional rectilinear objects.

Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems. Break down complex problems into smaller tasks.

Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Using ICT Supplement the exercises using ICT. For example, use Guess the view and Building houses with side views from www.fi.uu.nl/wisweb/welcome_en.html. Given the side views, build the ‘house’. Rotate the house to see it from different perspectives. There are two levels, with 10 questions at each level.

Building houses with side views

Guess the view


Assessment


Each side of the regular hexagon ABCDEF is 10 cm long.

Find the length of the diagonal AC.

TIMSS Grade 12

A stone monument consists of a square pedestal of edge 3 metres and height 1 metre topped by a cone of radius 1.5 metres and height 3 metres so that its base just fits the top of the pedestal. Use DGS to draw the plan and elevation of the figure with respect to a baseline that makes an angle of 30° with an edge of the base.

AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle (other than A or B), and S is the centre of the circle inscribed into ABC.

Then the measure of:

A. ∠ASB changes as C moves on k.

B. ∠ASB is the same for all positions of C but it cannot be determined without knowing the radius.

C. ∠ASB = 135° for all C.

D. ∠ASB = 150° for all C.

TIMSS Grade 12

Assessment


Use DGS to construct:

• a circle centre O;

• an equilateral triangle OAA′, two sides of which are radii of the circle;

• two further such equilateral triangles OBB′, OCC′ using different radii (which may or may not overlap);

• the lines A′B, B′C, C′A and their corresponding mid-points P, Q, R.

What kind of triangle is PQR?

Demonstrate that the result generalises to the case of three equilateral triangles with a common point but which are not necessarily congruent.

Unit 12F.2

225 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.3 | Statistics © Education Institute 2005

GRADE 12F: Statistics

Working with data

About this unit This unit is the first of two on statistics and probability for Grade 12 foundation. It covers the elements of sampling and data analysis. Much of it consolidates work begun in Grade 11 foundation.





Previous learning To meet the expectations of this unit, students should already be able to work with samples, have some experience of questionnaires, be able to calculate and use measures of central tendency and spread, and be familiar with techniques of data representation, such as histograms.

Expectations By the end of the unit, students will solve routine and non-routine problems in a range of mathematical and other contexts, and will use mathematics to model and predict the outcomes of real-world applications. They will break down complex problems into smaller tasks. They will approach problems systematically, knowing when it is important to enumerate all outcomes. They will conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They will synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They will explain their reasoning, both orally and in writing. They will recognise when to use ICT, and do so efficiently. They will arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations. They will use secondary data from published sources, including the Internet. They will use ICT to calculate statistical quantities and to produce a range of graphs, charts and tables to present and justify their findings. They will calculate measures of spread, including the variance and standard deviation. They will construct histograms and plot cumulative frequency distributions, using grouped continuous data if necessary. They will use simple simulations and consider trends over time using a moving average.

Students who progress further will have an increased range of techniques and experience from which to make appropriate choices when solving problems.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • computers with spreadsheet software and Internet access for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • histogram, frequency, (cumulative) frequency distribution, frequency

density, relative frequency, relative frequency distribution, time series, historigram, trend, seasonal variation, cyclical variation, residual variation, line of best fit, stem-and-leaf diagram, stem plot, box-and-whisker plot, box plot

• range, percentile, interquartile range, semi-interquartile range • mode, modal class, modal frequency, mean, moving average • variance, standard deviation • variation, population, random sample, stratified sample, representative

sample • questionnaire, survey, primary data, secondary data, discrete data,

continuous data, categorical data • hypothesis • bias

UNIT 12F.3 11 hours



11 hours SUPPORTING STANDARDS Grade 11F standards

CORE STANDARDS Grade 12F standards

EXTENSION STANDARDS Grade 12AQ standards


12F.1.2 Use mathematics to model and predict the outcomes of substantial real-world applications; compare and contrast two or more given models of a particular situation.


12F.1.7 Explain their reasoning, both orally and in writing.

12F.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.

12F.1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.


12F.1.14 Recognise when to use ICT and when not to, and use it efficiently.

11F.8.1 Know that: • it is important to choose representative

samples; • in a random sample there are chance

variations; • in a biased sample there are systematic

differences between the sample and the population from which it is drawn;

and locate obvious sources of bias within a sample.

12F.10.1 Know that: • it is important to choose representative samples; • in a random sample there are chance variations; • in a biased sample there are systematic differences between the

sample and the population from which it is drawn; and locate obvious sources of bias within a sample.

12AQ.9.1 Know the difference between categorical data, discrete data and continuous data.

2 hours

Sampling

2 hours

Questionnaires

2 hours

Data representation and measures

2 hours

Moving averages

3 hours

Statistical project

11F.8.2 12F.10.2 Plan surveys and design questionnaires to collect meaningful primary data from samples (including data collected in other subjects, such as science, geography or history) in order to make estimates of, or test hypotheses about, quantities or attributes characteristic of the population as a whole.

12AQ.10.4

Plan surveys and design questionnaires to collect meaningful primary data from samples in order to test hypotheses about or estimate characteristics of the population as a whole; formulate problems using secondary data from published sources, including the Internet.

12F.10.3 Formulate problems using secondary data from published sources, including the Internet.

Distinguish between population, sample and census; know the importance of choosing a representative sample; locate obvious sources of bias within a sample.

11F.8.3 Calculate and use measures of central tendency such as the arithmetic mean and the median.

12F.10.4 Calculate measures of central tendency such as the arithmetic mean and the median.

Unit 12F.3





12F.10.5 Calculate measures of spread, including the variance and standard deviation.

12AQ.9.14 Calculate measures of spread, including the variance and standard deviation; know the distinction between population and sample variance, and the corresponding standard deviations.

11F.8.4 Construct (relative frequency) histograms and plot cumulative frequency distributions, grouping continuous data when necessary.

12F.10.6 Construct (relative frequency) histograms and know that the area of each block of the histogram represents the frequency of occurrence of the respective class interval associated with the block; plot cumulative frequency distributions, using grouped continuous data if necessary.

12AQ.9.19 Interpret, in qualitative terms, the skewness of a frequency distribution and understand the importance of a symmetric distribution.

11F.8.6 Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings.

12F.10.7 Make inferences and arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings.

12F.12.1 Consider trends over time and calculate a moving average. 11F.9.1 Use a calculator with statistical functions to

aid the analysis of large data sets, and ICT applications to present statistical tables and graphs.

12F.14.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.


Activities


2 hours

Sampling Know that: • it is important to choose

representative samples; • in a random sample there

are chance variations; • in a biased sample there

are systematic differences between the sample and the population from which it is drawn;

and locate obvious sources of bias within a sample.

Explain their reasoning, both orally and in writing.

Sampling Remind students of their work on estimating the number of dots in a grid from Grade 11F. Use this to remind them: • why sampling is necessary; • the types of sample that can be drawn.

The two main types of sample are random and stratified. Give some examples from a school or college situation. • If some sort of survey is to be conducted, it will be very time-consuming to ask every student

in a school or college; the information needed could probably be obtained by restricting the survey to a sample only.

• If the survey concerns diet or size of family, for example, a random sample would suffice. • If the survey concerns shoe sizes or some other question for which the response is clearly

likely to be age-dependent, a stratified sample (where students are stratified by grades, for example) would be more appropriate.

Encourage students to think out why samples may be biased, and try to tie these to their own instincts and situations. They need to see the potential issues and to link them to possible techniques to exploit them.

Emphasise the need to design a sampling process carefully before going out to collect data.

A random sampling process is one in which each possible sample of the specified size is equally likely. This definition has some unexpected consequences. For example, a school has 100 students listed alphabetically; a sample of 10 is drawn by using a random number between 1 and 10 for the selection of a first one from the list, then every 10th student thereafter is selected. Although this process is obviously free from bias, it is not random since there are actually only ten possible samples that will result, whereas there are many more (100C10) to make up all possible samples.

There is a summary of the different types of sample on home.xnet.com/~fidler/triton/math/ review/mat170/samp/samp2.htm.


Project start Ask students to work in groups to formulate an area of investigation relating to everyday life, and to consider what sort of sampling process could be used to investigate it. They should formulate a clear hypothesis and summarise their findings.

Topics to consider are: • students’ leisure activities; • parents’ television viewing habits; • means of travel to school or work; • students’ shoe sizes.

Ask students to prepare their sampling methods by considering: • what sort of sampling process is appropriate; • how the sampling would be carried out (precisely).

Ask students to share and discuss their plans, and to assess the sampling methods for their appropriateness and practicability.

There is a list of suitable project material and sources in Unit 11F.5. It may be appropriate to allow students to rework existing projects to a greater degree of sophistication.

On the web There is a resource for statistical diagrams at www.shodor.org/interactivate/activities/tools.html.

Boxplot tool from Shodor

Unit 12F.3



2 hours

Questionnaires Plan surveys and design questionnaires to collect meaningful primary data from samples (including data collected in other subjects, such as science, geography or history) in order to make estimates of, or test hypotheses about, quantities or attributes characteristic of the population as a whole.

Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.


Approach a problem systematically, recognising when it is important to enumerate all outcomes.

Designing a questionnaire Remind students of lessons learned from previous experience of questionnaires, and elaborate on them as necessary. The main rules are: • never ask a leading question designed to get a particular response; • never ask irrelevant questions; • avoid personal questions unless they are directly relevant to the survey; • make every question as simple as possible; • make sure that each question will get a response from everyone to whom it is put.

Ask students to discuss in groups suitable questions for a survey on a given topic. Take feedback and discuss their ideas. Invite other groups to give constructive criticism of each group’s proposals.

Discuss the advantages and disadvantages of questions with a given selection of answers: • they make subsequent processing easier; • the responses must include all possible ones, including a ‘none of these’ option if needed.

Examples of questions to consider are: • Which age group are you?

(with categories listed – say, under 20, 21–30, 31–40, etc., over 70) • How many brothers and sisters do you have? • Do you like to try out new food?

Continue by: • posing an issue for consideration (such as a general question of diet or lifestyle); • seeking a response from students (purpose of the enquiry, hypothesis to test, and so on); • setting the task of designing a questionnaire in groups; • sharing and discussing the responses.

Examples of poor questions • How old are you? (personal) • Do you like sport? (too open-ended) • Do you agree that governments should never

give in to terrorism? (posed to expect agreement and begging the question of just what the situation may be)

• Do you try the food of the country when you go abroad? (can be answered only by those who have been abroad)

• If someone makes you a gift of clothing or of food, do you feel obliged to wear it or eat it under all circumstances? (far too complicated)

Project continued Ask students to develop the survey methods they designed in groups by proposing appropriate survey questions. When they have done this, ask them to share their ideas and to criticise the questions.

Take one or more of their ideas through to a conclusion by carrying out the survey and writing up the conclusions.



2 hours

Data representation and measures Construct (relative frequency) histograms and know that the area of each block of the histogram represents the frequency of occurrence of the respective class interval associated with the block; plot cumulative frequency distributions, using grouped continuous data if necessary.

Calculate measures of central tendency such as the arithmetic mean and the median.

Calculate measures of spread, including the variance and standard deviation.


Recognise when to use ICT and when not to, and use it efficiently.

Representation of data Students will have an understanding of the nature of statistics from their previous studies, as well as from general knowledge. Begin by trying to draw out their understanding of various mathematical issues. These are: • categorical and numerical data; • discrete and continuous numeric data; • grouped numeric data.

Check that students know how to represent categorical data (bar charts, pie charts). Check also that they can handle grouped data, and revise the reasons for it.

Make sure that students are already familiar with and have constructed on paper the types of graphs, charts and tables that they generate with ICT. Projects Ask students to select two or more subsets of the Mayfield High School data (boys’ heights and girls’ heights, for example) for comparison. Use ICT to represent the data by: • histograms using suitable class intervals; • cumulative frequency diagrams.

Ask students to use the cumulative frequency diagrams to estimate the medians and interquartile ranges for each subset. Ask them to write a sentence to summarise what these diagrams and measures show.

Ask groups to exchange their reports and to scrutinise each other’s work. This will give an opportunity to check that they can turn histograms back into approximate frequencies.

Use this to motivate discussion of standard deviation and variance. Explain briefly how these work and how to use both computers and calculators to find their values.

Ask students to use their work to compare and contrast the calculation of: • mean with median; • semi-interquartile range with standard deviation.

Sources of data Several data sources can be tapped for this work. There is Qatar data at www.census.gov/cgi-bin/ipc/idbsum.pl?cty=QA. Other sources are given in Unit 11F.5.

The Oundle-TSM site has a list of sources at www.tsm-resources.com/mlink.html#stats (see section 4, Probability and statistics).

Mayfield High School is another suitable large data set available at www.edexcel.org.uk/virtualcontent/17838.xls.



2 hours

Moving averages Consider trends over time and calculate a moving average.

Use mathematics to model and predict the outcomes of substantial real-world applications; compare and contrast two or more given models of a particular situation.

Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.


Time series

Discussion Present the class with (say) sales data of a particular item at a business by year. Explain that this is an example of a time series: a series of observations of a variable taken at intervals in time. Ask students how they would represent this; obvious answers are graphs, bar charts, histograms.

Focus on graphical representation – sales plotted against time, for example. Say that this is sometimes called a historigram (not the same as histogram).

Explain the terms: • trend: the basic underlying movement with other fluctuations smoothed out; • seasonal variations: short-term regular fluctuations about the trend, such as may be evident in

any particular year’s breakdown by months; • cyclical variations: long-term fluctuations about the trend, corresponding to highs and lows in

economic activity (evident in the graph on the right where there is a clear five-year cycle); • residual variation, which is caused by unusual events such as a serious fire to business

premises.

Explain how to estimate trend: • the simplest technique is a line of best fit, which is applicable when there are no cyclical or

seasonal variations; • the method of moving averages is available when there are cyclical variations.

Explain how to calculate a moving average by estimating the length of a cycle (five years in the graph in the notes) and working out an average for each consecutive set of five figures. Index these averages by the mid-value of the series of five, as in the second graph. The moving averages now are the basis of a trend graph. Explain that a line of best fit can be used to extend the graph into the future to form the basis of prediction.

Point out to students that different styles of line are important to distinguish the different models here, to distinguish the trend from the historical, and to distinguish predictions from data values.

Go on to discuss different ways of looking at such data. When considering sales, for example, the model to use is: sales = trend + seasonal variation + cyclical variation + residual variation

This will work when the trend is changing slowly, or over the short term.

Examples of time series data: • birth dates (by week); • unemployment/employment figures (monthly); • share index numbers (published daily for the

international stock markets).

A typical sales graph indexed by years

The same graph with a moving-average graph added



Moving averages in Excel In the extract from a spreadsheet on the right, based on Microsoft Excel, the first column gives the year and quarter for which the sales appear in the second column.

Excel’s chart drawing facility gives the chart below, and it has added a moving average based on a period of four quarters. (Excel plots the moving average is plotted over the right-hand end-point of the period rather than over the more conventional mid-point.) Explain this to students so that they can use it as a tool.

Example Find out about the cost, in Qatar, of a barrel of crude oil over the period January 2000 to March 2004. Analyse the data over periods of three months and consider the moving-average price per barrel. Discuss your findings.

Project For a period of four weeks: • note the exchange rate between two major world currencies (e.g. US dollars and pounds

sterling); • plot a line graph of the data; • estimate the trend and predict the exchange rate for each day in the following week; • compare those estimates with the actual values when they become available.



3 hours

Statistical project Formulate problems using secondary data from published sources, including the Internet.

Make inferences and arrive at conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings.



Use mathematics to model and predict the outcomes of substantial real-world applications; compare and contrast two or more given models of a situation.

Break down complex problems into smaller tasks.



Working with real data

Project and assessment The objective is to collect data by sampling from a secondary data source and to analyse it to answer a question. The challenge is to motivate students to plan properly. Planning begins with questions and hypotheses that are well formulated – and, if possible, interesting to the student.

Have ready a fallback option to use as an exercise for students or as an example to motivate them, or possibly a bit of both, if necessary. However, expect students to formulate questions, find the data and carry out the sampling on their own initiative. Monitor each stage carefully: at the outset, for example, get students to write down and submit for vetting just what they propose to do, since this helps them to clarify their thoughts and also allows for your intervention if needed.

Students should: • formulate question(s); • propose data sources from which to find the answers; • plan a sampling process and analysis so that the question(s) may be satisfactorily answered; • write up their work so that all these stages of development are clearly presented; • state their conclusions clearly.

Remind students (if necessary) of the various charts they can use (bar chart, pie chart, stem-and-leaf diagram, histogram, cumulative frequency diagram, box-and-whisker plot). Stress that it is unnecessary to show the same information in several different ways unless significantly different elements are emphasised; it is better rather to choose one and give the reasons why. Advise students also that, in their written report, diagrams and tables are much easier to produce and take in than prose, but that all diagrams should be accompanied by appropriate, brief explanations.

Students will need time to do this work. If this is their first such project, they will use a lot of time at first apparently going nowhere. Frequent intervention with restrained advice will be essential, as well as a clearly understood timeframe in which to carry out the project and complete it.

Before they begin, give students the criteria you intend to use to assess their work; they can use the criteria as a checklist as they complete the work. Suitable criteria are: • Is the student’s objective clear? Is each question clearly posed? • Is the data source adequate for the purpose? • Is the sampling process: (a) clear; (b) appropriate; (c) correctly carried out? • Is the analysis clear and accurate? • Is the conclusion clear and correct (within the scope of the project)?

It is appropriate for students to work in groups to plan and collect data to be pooled and shared; they may need guidance on how to explain this in their work.

The underlying principles of mathematical modelling should underpin the project work. Students should see that every exercise of this sort is essentially unfinished, since every investigation leads inevitably to new questions. Encourage students to write in their conclusion what they consider should be done next.

See the earlier list of resources for data sets.

Rainfall data, for example, is available at www.waterplc.com/WaterPlc/news/stats/ rain01.html (see the zip file at the foot of that page).

Suitable rainfall questions are: • Compare and contrast rainfalls in the month of

March in two successive years (very simple and largely descriptive).

• Take two ten-year periods and decide whether there was a significant difference in rainfall pattern from one to the other (which will require some thought about how to characterise the rainfall patterns, possibly in several different ways).

Other project options and sources are listed in Unit 11F.11.


Assessment


Find the mean and median salaries of the group of workers in Qatar whose weekly salaries in riyals are given in the table on the right.

Salary 250 300 350 400 450 500 550 600

Frequency 5 11 20 31 18 12 7 3

Assessment


The table below shows the number of cars leaving a car park during the periods given.

Minutes after 1700 h 0 ≤ n < 5 5 ≤ n < 10 10 ≤ n < 20 20 ≤ n < 50 50 ≤ n < 60

Number of cars leaving 74 115 248 1174 189

Complete the histogram on the right to show the information in the table.

Write the frequency density above each rectangle of the histogram.

The value 14.8 on the histogram is the frequency density for the period 0 ≤ n < 5 minutes. Explain what is meant by frequency density with regard to cars leaving the car park.

The following table shows the sales of a product during the years 1999–2007.

a. Draw a historigram of these data.

b. Comment briefly on the types of variation present in this time series.

c. Calculate moving averages over an appropriate period and plot these on your graph. Draw a trend line by eye.

Estimate the trend value for 2007 and hence estimate the actual sales in 2008.

Year 1999 2000 2001 2002 2003 2004 2005 2006 2007

Sales (000’s)

448 365 465 466 392 483 493 413 510

Sulaiman did a survey of the age distribution of 160 people at a theme park. The cumulative frequency graph shows his results.

a. Use the graph to estimate the median age of the 160 people at the theme park.

b. Use the graph to estimate the interquartile range of the age of the 160 people at the theme park.

Unit 12F.3

235 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.4 | Functions 1 © Education Institute 2005

GRADE 12F: Functions 1

Functions, equations and graphs

About this unit This unit is the second of five on algebra in Grade 12 foundation. The unit offers more work with functions and relates their algebraic form to their geometric representation.





Previous learning To meet the expectations of this unit, students should already be able to use a graphics calculator to display graphs of functions and interpret them. They should also distinguish functions from relations and know how this distinction is made graphically.

Expectations By the end of the unit, students will identify and use connections between mathematical topics. They will perform appropriate manipulations and calculations. They will recognise when to use ICT and do so efficiently. They will generate further formulae from physical contexts. They will solve a range of problems using inverse functions. They will use realistic data and ICT to analyse problems.

Students who progress further will have gained greater facility in abstract language.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • graph plotting software such as:

Autograph (see www.autograph-math.com) Graphmatica (free from www8.pair.com/ksoft)

• computers with spreadsheet and graph plotting software for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • expansion, factor, multiple, divisor • inverse, symmetry, domain, range, odd, even, periodic, function, mapping,

one-to-one, relation • equation, linear, quadratic, quadratic formula

UNIT 12F.4 12 hours









11F.5.1 Use a graphics calculator to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts.

12F.5.1 Use a graphics calculator, including the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

11F.5.9 Recognise a second-order polynomial in one variable, y = ax2 + bx + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas), and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when such functions are increasing, when they are decreasing, and when they are stationary.

12F.5.2 Use physical contexts to plot and interpret: • graphs of linear, quadratic and cubic functions; • graphs of the reciprocal function y = k / x (x ≠ 0); • graphs of the sine and cosine functions; • graphs of the modulus function and a range of simple

non-standard functions.

12AQ.5.3 Understand the modulus function y = | x | and sketch its graph.

11F.5.15 Solve physical problems modelled simultaneously by two such functions.

2 hours

Properties of curves

4 hours

Properties of graphs 2 hours

Inverse functions

2 hours

Combinations of functions

2 hours

Using a graphics calculator to solve simultaneous equations

11F.5.6 Know that a straight line in the explicit form y = mx + c represents a function, but that a straight line in the implicit form ax + by + d = 0 may, or may not, be a function; know that any straight line in the xy-plane can be represented in this implicit form, but that only certain lines in the plane can be represented by the explicit form; work with both of these forms.

12F.5.3 Find, graph and use the inverse function of those functions in 12F.5.2 given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.

12AQ.5.512AS.5.5

Form inverse functions (on a restricted domain, if necessary) and use the notation y = f–1(x).

11F.5.2 Recognise when a graph represents a functional relationship between two variables and when it does not.

12F.5.4 Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

Unit 12F.4


Activities


Investigating symmetries

Investigation 1 Use a graphics calculator (or Autograph) to graph the functions:

• 1yx

=

• 2yx

=

• kyx

= (for other values of k, including negatives)

• What symmetries do these graphs have?

Investigation 2 Use a graphics calculator (or Autograph) to graph the functions: • y = x2 • y = 3x2 + 2 • y = ax2 + c (for other values of a and c, including negatives) • What symmetries do these graphs have?

Extend your investigation to the graphs y = ax2 + bx + c.

Using Autograph to investigate kyx

=

There is a longer and more detailed investigation into quadratics in Unit 10A.10. This may be suitable for students who are growing in confidence about this topic.


2 hours

Properties of curves Identify and use interconnections between mathematical topics.

Use physical contexts to plot and interpret: • graphs of linear, quadratic

and cubic functions; • graphs of the reciprocal

function y = k / x (x ≠ 0); • graphs of the sine and

cosine functions; • graphs of the modulus

function and a range of simple non-standard functions.

Recognise when to use ICT and when not to, and use it efficiently. Investigation 3

Use a graphics calculator (or Autograph) to graph the functions: • y = x3 • y = x3 + 2 • y = x3 + 3x2 + 4x + 2 • y = ax3 + bx2 + cx + d (for other values of a, b, c and d, including negatives) • What symmetries do these graphs have?

Investigation 4 Use a graphics calculator (or Autograph) to graph the functions: • y = sin x • y = sin kx (for other values of k) • y = cos kx • What symmetries do these graphs have?

Unit 12F.4



4 hours

Properties of graphs Identify and use interconnections between mathematical topics.

Use physical contexts to plot and interpret: • graphs of linear, quadratic

and cubic functions; • graphs of the reciprocal

function y = k / x (x ≠ 0); • graphs of the sine and

cosine functions; • graphs of the modulus

function and a range of simple non-standard functions.

Odd, even and other types of function and graph

Discussion Begin by defining the different types of symmetry of a graph y = f(x) and the words associated with them. These are: • odd functions, where f(–x) = –f(x); • even functions, where f(–x) = f(x).

Illustrate with examples from the functions that students have met – parabolas, cubics and the reciprocals. Discuss how these correspond to geometric symmetries: • odd functions: the graph has rotation symmetry of order 2 about the origin; • even functions: the graph has reflection symmetry in the y-axis.

Introduce at least one new function. An example is the modulus function:

f(x) = | x | = when 0 when 0

x xx x

≥⎧⎨− <⎩

Discuss how to draw the graph of the function and highlight its principal features.

Note that the case of x = 0 may provoke some discussion: it can be included in the first line or the second of the definition. Mention also: f(x) = [ x ] (or INT[x]) (the integral part of x, so [ 3.6 ] = 3)

On the web This topic is explored further but accessibly at staff.imsa.edu/math/journal/volume4/articles/ ExploreEvenOdd.pdf.

Challenge Use the digit 2 exactly four times to find expressions, using any functions and operations from your experience, that will produce each of the integers from 1 to 20.

First answers:

• 2 212 2

×=×

• 2 222

2 −=

• 23 2 22

= × −

Practice Give students examples to practise. Introduce further new functions as they work on exercises. Set questions that combine pencil-and-paper sketching with investigations on graphics calculators. Use applets to extend students’ experience.

Further ideas to develop through these are: • sine and cosine functions over extended ranges, and their properties; • periodic functions; • increasing and decreasing functions; • first acquaintance with the functions y = ln x and y = ex.

Examples • Sketch the graphs of the functions y = | x – 1 | and y = | x – 1 | + | x – 3 |. • Sketch the graphs of the functions y = sin [ πx – π ] and y = | x |2 + 3 | x | + 2. • Find the domain over which y = x2 – 5x + 4 is an increasing function. • Solve the equation | x + 1 | = x2 – 5x + 4.

Introduce physical contexts whenever possible. At the end of this activity, summarise the key points and definitions for future reference.

Use the applets from www.fi.uu.nl/wisweb/welcome_en.htm:

Find the function Scope

Playing with functions



Finding an inverse function

Discussion

Pose the simple problem of making t the subject of the formula 5 7 ,3

tx += giving 3 7 .5

xt −=

Then ask students to reconsider the problem from the point of view of functions and graphs. The arrow diagram (in the notes column) shows the functional relationship as a rule to map the domain elements {1, 4, 7, 10} to the range elements as shown. The same diagram drawn from right to left (or with the arrows reversed) represents the inverse function derived in a similar way.

The graph shows the two functions as 5 7 3 7, ;3 5

x xy y+ −= = note that the letters have

changed but that the functions are unchanged. Ask students how the two graphs are related.

From these diagrams and the algebra, explain what is meant by inverse function, establish how to derive an inverse and confirm that the two graphs are related by reflection in y = x.

Now extend the concept of inverse to more complex situations. Return to the distinction between graphs that represent functions (or mappings) and those that do not. Find out if students remember how this is obvious from a graph. Convince them from this that an inverse (function) exists only if a function is one-to-one on its domain of definition. Explain how the domain can sometimes be restricted to make this so. (For example, y = x2 has an inverse if the domain is restricted to x ≥ 0.)

1 → 4

4 → 9

7 → 14

10 → 19

5 7 3 7, ,

3 5x xy y y x+ −= = =

2 hours

Inverse functions

Find, graph and use the inverse function of those functions in 12F.5.2 given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.


Exercises Set questions that cover: • revision of graphs to distinguish functions from relations; • determination of inverses of one-to-one functions; • sketches of functions and their inverses; • functions that can be considered one-to-one by suitably restricting their domains; • inverses of functions with restricted domains.



2 hours

Combinations of functions

Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

Combining functions

Class discussion Explain that functions can be added, subtracted, divided and multiplied just like numbers. Give examples in the form of polynomials that are combinations of powers, and of brackets that multiply out to give equivalent expressions (identities).

Exercises Give students exercises on these procedures.

Example If f(x) = x3 – 1, g(x) = x2 + x + 1 and h(x) = x – 1, find and simplify: • f(x) + g(x); • f(x) – g(x)h(x);

• f( ) g( ) .f( ) h( )

x xx x

+−

On the web MathsNet has pages devoted to functions at www.mathsnet.net/asa2/2004/c3.html#1.

Another good source is at Visual Calculus archives.math.utk.edu/visual.calculus/0/ compositions.5/index.html.

Extension: long multiplication This is covered in depth in Unit 10A.3.

Here is an example of long multiplication taken from Unit 12AQ.1.

x xx x

x xx x x

x x xx x x x

− ++ −

− + −− +

− +− − + −

2

2

2

3 2

4 3 2

4 3 2

2 5 35 7 2

4 10 614 35 21

10 25 1510 11 24 31 6

2 hours

Using a graphics calculator to solve simultaneous equations



[continued]

Solving simultaneous equations with graphics calculators

Class discussion Students have considered simultaneous equations from an algebraic point of view and have related the algebra to the corresponding graphs. This section brings these ideas together and exploits the graphics calculator.

Begin by discussing an example. • Obtain a graphics calculator display of the curves: y = 5 + 6x – 3x2 – x3

y = x3 – 4x + 1 • Use the zoom function to obtain estimates (to one decimal place) of the x-coordinates of the

intersection points. • Hence solve the inequations: 5 + 6x – 3x2 – x3 > x3 – 4x + 1; 0 > 4 + 10x – 3x2 – 2x3.



[continued] Use a graphics calculator, including use of the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

Exercises Set questions of the same sort as the example.

In addition, introduce (using the graphics calculator) the graphs of the exponential and logarithmic functions.

Use physical contexts (such as growth and decay situations) to create problems that pose the same type of question. The trace function can be used to relate the solutions to the graphs.

Using ICT Use the applet Growth from www.fi.uu.nl/wisweb/welcome_en.html.


Assessment


Which grows faster for x ≥ 0: the power function y = x3 or the exponential function y = ex?

Justify your answer.

If xy = 1 and x is greater than 0, which of the following statements is true?

A. When x is greater than 1, y is negative.

B. When x is greater than 1, y is greater than 1.

C. When x is less than 1, y is less than 1.

D. As x increases, y increases.

E. As x increases, y decreases.

TIMSS Grade 12

Investigate physical examples of inverse square laws.

Find physical examples that are modelled by circular functions.

A big wheel makes one complete revolution every 90 seconds. The wheel has a diameter of 20 metres. The bottom of the wheel is 2 metres above the ground. Two people get on the wheel and sit in a seat, and then the wheel starts to rotate. T seconds later their height above the ground is given by h = 2 + 8 sin 4T °.

Explain why this is an appropriate formula to use.

At what two consecutive times are the people 12 m above the ground?

Assessment


A ship can enter a harbour only when the tide is in; it must have a minimum depth of water of 8 metres. The tide follows a daily sinusoidal variation given by the formula d = 5 sin 15t ° + 8, where t is the time in hours from midnight onwards, measured on the 24-hour clock.

At how many times in a day will the depth of water in the harbour be exactly 8 m? For how many hours a day can the ship enter the harbour?

Sketch how the level of the tide varies with the time of the day.

The cost of production of q silver bracelets is C = 200 + 15q. Find the inverse function and interpret its meaning.

Calculate the inverse function of f(x) = 5x – 8.

Unit 12F.4


GRADE 12F: Geometry 2

Enlargements, plans and compositions

About this unit This unit is the second of two on geometry for Grade 12 foundation. It covers transformations and practical geometry.





Previous learning To meet the expectations of this unit, students should already be able to use a dynamic geometry system (DGS) to carry out elementary transformations (as covered in Grade 9).

Expectations By the end of the unit, students will solve routine and non-routine problems in a range of mathematical and other contexts, and use mathematics to model and predict the outcomes of real-world applications. They will break down complex problems into smaller tasks, and set up and perform appropriate manipulations and calculations. They will conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They will synthesise, present, interpret and criticise mathematical information, working to expected degrees of accuracy. They will recognise when to use ICT and do so efficiently. They will interpret maps and scale drawings, and translate, reflect, rotate and enlarge two-dimensional geometric objects. They will explore aspects of geometry using ICT.

Students who progress further will gain a deeper grasp of the way transformations combine.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access and computer linked to data projector • dynamic geometry system (DGS) such as:


• word processing software such as Microsoft Word • presentation software such as Microsoft PowerPoint • computers with dynamic geometry system, word processing and

presentation software for students • digital cameras (optional)

Key vocabulary and technical terms Students should understand, use and spell correctly: • transformation • reflection, axis of reflection • rotation, angle of rotation, centre of rotation • translation • enlargement, centre of enlargement, scale factor • composition of transformations • plan, scale • commutative transformations

UNIT 12F.5 7 hours




Grade 9 standards CORE STANDARDS Grade 12F standards


12F.1.1






2 hours

Harder enlargements

3 hours

Scale drawings

2 hours

Composition of transformations

12F.1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.


12F.1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.


12F.7.1 Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line; understand the meanings of positive, negative and fractional scale factors in enlargements.

9.5.6 Identify a single transformation mapping a 2-D shape onto its image: reflection, rotation, translation or enlargement by a positive integer scale factor; find a line of reflection, centre or angle of rotation, scale factor or centre of enlargement in simple cases.

12F.7.2 Interpret maps and scale drawings.

9.5.7 Identify and draw, on paper and using ICT, the enlargement of a simple plane figure by a positive fractional scale factor; identify the scale factor as the ratio of two corresponding line segments.

12F.7.3 Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle. (See also 12F.7.1.)

Unit 12F.5


Activities


2 hours

Harder enlargements Understand the meanings of positive, negative and fractional scale factors in enlargements.



Working with enlargements

Investigation 1: enlargements explored further Use a dynamic geometry system (DGS) for this. • Draw triangle XYZ where X has coordinates (1, 1), Y (–2, 3), Z (3, 3). • Start with the original XYZ each time and show the figure enlarged with centre (0, 0) and

scale factors 3, 0.5, –2. • Repeat with centre (2, 1) and the same scale factors. • Note any special features. • Investigate the transformation needed to return each image figure to the original XYZ.

Can you make any generalisation about how to do that? • Work with other figures, centres and scales to test your conclusions.

Discussion Ask students for their ideas on enlargements. The responses will include some or all of: • angles unchanged; • sides in proportion; • centre and scale factor to be prescribed.

Ask students if there are any special cases or limitations on the value of k, the scale factor. Special cases would be: • k = 0 (all points map to the origin); • k = 1 (identity transformation); • k = –1 (reflection in the origin).

Move on to consider k negative. Show how the enlargement is constructed in this case for (say) a triangle. Ask students how the result differs from what they have seen before. They may mention that the image is oriented differently.

Demonstrating enlargement with Sketchpad


Investigation 2: working it out Do these either on squared paper or using DGS. • A (–1, 1), B (3, 1), C (3, 3) and D (–1, 3) are the vertices of a rectangle. P is the point (2, 0).

Draw ABCD, and its image A1B1C1D1 under the enlargement [P, –3], where P is the centre of enlargement and –3 is the scale factor. Write down the coordinates of A1, B1, C1 and D1.

• List the coordinates of six points on the line y = 2x + 4. List the coordinates of their images under [O, 3], where O is the origin. Find the equation of the line on which the images lie.

Practice using ICT Use Transformations – Dilation (nlvm.usu.edu/en/nav/vlibrary.html).

Unit 12F.5



3 hours

Scale drawings Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

Interpret maps and scale drawings.


Work to expected degrees of accuracy, and know when an exact solution is appropriate.

Use a range of strategies to solve problems.

Mapping project Make a scale drawing for a practical purpose. For example, plan the use of an athletics ground or sports field for a major event, such as an athletics meeting or football match. An alternative would be the use of the school land if there were no buildings on it and it were redeveloped for alternative use. The plan should begin with: • purpose and requirements; • consideration of the best scale; • surveying method; • ways of arranging the various elements: areas for track and field events, for example.

Students should work on this project in groups. At an early stage, when each group has prepared a plan for the execution of the work, groups should compare notes to discuss their different approaches, so that the best ideas can emerge.

Ask students to complete the work to a deadline and to produce it in a suitably polished form, with ICT. This could include word-processed text with diagrams inserted and photographs taken with digital cameras.

To assess this, allow for: • a clear statement of purpose and design intentions; • quality and accuracy of plans; • faithfulness to the design; • presentation and summary of advice and conclusions; • application of ICT skills.

Students could prepare a presentation on their project using Microsoft PowerPoint, inserting diagrams and photographs.

Dimensions of areas for athletics are available at www.ausport.gov.au/fulltext/1998/wa/dimplay.asp (for example).

Al Saad football stadium

The time for this project could be extended by a few hours to allow for the further application of ICT skills.



2 hours

Composition of transformations Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.


Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line.

Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle.


Successive transformations

Investigation Use DGS or Autograph as the basis of an investigation into combinations of transformations. Here is an example of how to structure it. • Reflection in two parallel lines: – Draw the points (1, 3), (2, 1), (4, 2) of a triangle T. Draw the line x = –1. – Reflect triangle T in the line x = –1 and call the result U. – Reflect U in the line x = 0 and call the result V. – How is result V related to triangle T? • Repeat for three other points of your choice, and two other parallel lines. What generalisation

can you make from your conclusions? • Repeat the investigation on triangle T, but this time with successive reflections in x = –1 and

y = 2. • Repeat with a triangle of points of your own choice and two non-parallel lines. What

generalisation can you make from your conclusions? • Repeat the investigations for two successive rotations. What generalisation can you make

from your conclusions? • Repeat the investigations for two successive enlargements about the same centre. What

conclusion do you draw? • Repeat the investigations for two successive rotations about a common centre. What

conclusion do you draw? • Try other combinations of transformations. Be careful to do so systematically.

Discussion From the results of the investigation, summarise the conclusions. Students should have found at least the following: • two reflections in parallel lines are equivalent to a translation; • two reflections in non-parallel lines are equivalent to a rotation about the point where the lines

intersect; • two successive enlargements with the same centre result in a single enlargement with their

scale factors multiplied together; • two rotations about the same point result in a single rotation with the sum of the two angles.

Two reflections in parallel lines in Autograph

The result is a translation (perpendicular to the lines and twice the distance between them).

Two reflections in non-parallel lines

The result is a rotation about the intersection of the reflection axes, of twice the angle between them.

On the web MathsNet has interactive pages on transformations. Start, for example, at www.mathsnet.net/transform/index.html.

There is also a page there on combining transformations. Its address is www.mathsnet.net/transform/comindex.html.

Ask students to distinguish the transformations that preserve congruence. To give a fully correct answer requires some care about special cases.

Summarise this by checking that students have realised that: • the image of a plane figure under rotation or reflection is congruent to the original figure

before rotation or reflection; • every circle is similar to any other circle.



Challenges • In ABC, ∠BAC = 40°. AB1C is the image of ABC under reflection in the line of AC;

AB1C1 is the image of AB1C under reflection in the line of AB1. – Describe exactly the single transformation which maps ABC to AB1C1. – How many such pairs of reflections will map ABC onto itself?

• OA, OB, OC and OD are four lines in that order intersecting at O. Let M1, M2, M3 and M4 represent the operations of reflection in OA, OB, OC and OD respectively. Describe the single operations equivalent to:

– M1M2 (i.e. M2 first, followed by M1); – M3M4; – (M1M2)(M3M4); – (M4M3)(M2M1); – (M3M4)(M1M2).

• What single transformation is equivalent to each of the following? – reflection in the x-axis followed by a quarter turn anticlockwise about the origin; – reflection in the y-axis followed by a quarter turn clockwise about the origin; – reflection in y = x followed by a half turn about the origin; – reflection in y = –x followed by a half turn about the origin.

More practice using ICT Use Transformations – Composition from nlvm.usu.edu/en/nav/vlibrary.html to explore the effect of applying a composition of translation, rotation and reflection transformations to objects.

• Which of these combinations is commutative? – two reflections; – two rotations; – two translations; – two enlargements.

The idea of commutative pairs of transformations is implicit in much of what has gone before. Able students will profit from meeting the concept and exploring it explicitly.


Assessment


An equilateral triangle ABC has side length 10 cm. It rotates around the inside of a square of side length 20 cm.

a. Triangle ABC rotates about C to the position shown as CA1B1. What is the angle of rotation?

b. Calculate the distance along the path travelled by point A in turning from A to A1.

c. Calculate the distance along the path travelled by point A in turning from A1 to A2.

d. The triangle continues rotating around the inside of the square in the same way until it is back at the original position. Which of the original points A, B or C will point A land on when it has completed its rotations around the inside of the square?

The rectangle Q in the diagram on the right CANNOT be obtained from the rectangle P by means of a:

A. reflection about an axis in the plane of the page

B. rotation in the plane of the page

C. translation

D. translation followed by a reflection

Circle the correct answer.

TIMSS Grade 12

A triangle has vertices at the points (4, 5), (6, 1) and (8, 11). The triangle is enlarged by a factor of 2 about a centre of enlargement at the point (3, –3). Draw the enlarged triangle in its correct position on a coordinate grid.

Assessment


Line segment OA is 3.0 cm long. Line segment OB is √ 7 cm long. OB can rotate in a horizontal plane about the point O.

Find the maximum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

Find the minimum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

Sketch a different position for line segment OB so that the distance from A to B, AB, is a rational number. Confirm by calculation that your answer is a rational number.

OB is reduced in length to 2.6 cm. OA is still 3.0 cm long. Calculate the distance AB when angle AOB is 120°.

The lengths of 2.6 cm and 3.0 cm are accurate to one decimal place. The 120° angle is accurate to the nearest degree. Calculate the greatest and least possible values of AB.

Unit 12F.5



The diagram shows parts of two circles, sector A and sector B.

a. Which sector has the bigger area?

b. The perimeter of a sector is made from two straight lines and an arc. Which sector has the bigger perimeter?

A semicircle, of radius 4 cm, has the same area as a complete circle of radius r cm. What is the radius of the complete circle?



Composite and inverse functions

About this unit This unit is the third of five on algebra for Grade 12 foundation. The unit focuses on composition of functions and extends work on inverses.





Previous learning To meet the expectations of this unit, students should already be able to manipulate elementary functions.

Expectations By the end of the unit, students will break down complex problems into smaller tasks and set up and perform appropriate manipulations and calculations. They will aim to generalise. They will solve a range of problems using inverse and composite functions.

Students who progress further will have seen function composition as a generalised form of multiplication.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector

Key vocabulary and technical terms Students should understand, use and spell correctly: • function, inverse, self-inverse, composition

UNIT 12F.6 6 hours









12F.5.5 Understand the concept of a composite function and use the notation y = f(g(x)).

4 hours

Composing functions

2 hours

Finding inverses of compositions

12F.3.1 Develop further confidence in all the calculation skills established in Grades 10 and 11.

12F.5.6 Deconstruct a composite function into its constituent functions, using inverse functions.

12F.4.1 Rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts.

Unit 12F.6


Activities


4 hours

Composing functions

Understand the concept of a composite function and use the notation y = f(g(x)). Break down complex problems into smaller tasks.


Composing functions

Preliminary exercise In each of the formulae below, make the substitution and simplify the result. • y = 3x + 2; x = t + 1 • y = 3x + 2; x = 2t • y = x2 + 2x – 2; x = t + 1 • y = x2 – 5x + 3; x = 2t – 3 • y = 3x3 + 2x2 – 4x – 2; x = 3t

Discussion Use the work on the preliminary exercise to establish the idea of composition, or a function of a function. There are two notations in use which bring out different aspects:

y = f(x), x = g(t) ⇒ y = f(g(t)) ≡ h(t), say

so we can write h = f ◦ g. (Note that this implies that the order of application of the functions in such a product is read from right to left.)

On the web MathsNet has pages devoted to functions (www.mathsnet.net/asa2/2004/c3.html#1).

Visual Calculus is another good source (archives.math.utk.edu/visual.calculus/0/ compositions.5/index.html)

Investigation Define six functions as follows:

0

1

2

3

4

5

f ( )1f ( )

f ( ) 11f ( )

11f ( ) 1

f ( )1

x x

xx

x x

xx

xxxxx

=

=

= −

=−

= −

=−

• Show that f1 o f5 = f4. • Show that f2 o f2 = f0. • Show that f5 o f2 ≠ f2 o f5. • Evaluate all other possible compositions of pairs and draw up a multiplication table to show your

answers.

The material in this section can be adapted at a variety of levels. In particular, although this is formulated as a challenge, the individual steps are excellent practice material on the basic composition concepts. Moreover, they naturally link in with inverses, since the multiplication table forms a closed system.

Model solution of first two examples (abbreviated):

1 5 41 1(f f )( ) 1 f ( )

1

x xx xx

= = − =⎛ ⎞⎜ ⎟−⎝ ⎠

o

( )2 2 0f f ( ) 1 (1 ) f ( )x x x x= − − = =o

See the next page for the multiplication table.

Unit 12F.6



Extension for able students Point out to able students that the table shows the properties of group structure: • no element occurs twice in any row or column; • f0 acts as an identity element; • each element has an inverse: that is, for element fn there is another fm such that their composition

(in either sense) is f0; • the associative law holds for function composition.

f0 f1 f2 f3 f4 f5

f0 f0 f1 f2 f3 f4 f5

f1 f1 f0 f3 f2 f5 f4

f2 f2 f4 f0 f5 f1 f3

f3 f3 f5 f1 f4 f0 f2

f4 f4 f2 f5 f0 f3 f1

f5 f5 f3 f4 f1 f2 f0

The left column is the first element of the product and the top row the second (so f2 o f4 = f1, for example).

Inverses again

Investigation continued • Find the inverse of each of the six functions (defined in the previous section). • Which function is self-inverse? Is there more than one possibility? • Draw up a table that pairs each function with its inverse.

What properties does this table possess? • Show that (f2 o f3)–1 = f3–1 o f2–1. Does this result generalise?

The generalisation question in this case is ambiguous: • the generalisation to the six functions

can be demonstrated exhaustively from the table;

• the general result for all compositions can be asserted too.

Proof of the second proposition above is a challenge for able students.

Working with compositions Move on to the idea of composition. Take an example such as y = sin (x3 + 1). Ask students what they would do first to evaluate such a function at a given value of x; give such a value if the response is slow. List the steps:

3 3 31 sin( 1)1 sin( 1)

sin

x x x xp p p

q q

⎯⎯→ ⎯⎯→ + ⎯⎯→ +⎯⎯→ + ⎯⎯→ +

⎯⎯→

Each of the steps is a function in itself, and the end result is the composition of them all.

3f

g

h1

sin

x xp p

q q

⎯⎯→⎯⎯→ +

⎯⎯→

So we have functional ‘steps’ which are defined by: p = x3 ≡ f(x), say q = p + 1 ≡ g(p) y = sin q ≡ h(q)

The function is finally expressed as y = h(g(f(x))).

This composition is used again in a later section of this unit.



Practice

Give students examples of this type to work on so that they can formalise the steps.

Example Decompose the function f(x) = (x2 + 2x + 2)3 + 1 into three or more functional ‘steps’.

2 hours

Finding inverses of compositions Deconstruct a composite function into its constituent functions, using inverse functions.


Working with inverses

The deconstruction method allows the inverse to be constructed by reversing the steps in the example used in the previous section. Thus:

1

1

11/ 3

h

g

f

arcsin1

y yr r

s s

−

−

−

←⎯⎯⎯− ←⎯⎯⎯

←⎯⎯⎯

That is: 1 1 11/ 3 f g h(arcsin 1) arcsin 1 arcsiny y y y

− − −− ←⎯⎯⎯ − ←⎯⎯⎯ ←⎯⎯⎯

So we have x = f–1(g–1(h–1(y))) = (arcsin y – 1)1/3.

Exercises Give students exercises on these procedures.

Examples • If f(x) = x3 – 1, g(x) = x2 + x + 1 and h(x) = x – 1, find and simplify: f(x) + g(x); f(x) – g(x)h(x);

f( ) g( ) .f( ) h( )x xx x

+−

• If f(x) = x3 and g(x) = x + 1, find and simplify f(g(x)) and g(f(x)). Solve the equation g(f(x)) = f(g(x)).


Assessment


h: x → 1 – 2x, k: x → x2 are two functions from to .

a. Find a formula for h o k and for k o h.

b. Find x for which (h o k)(x) = –7.

c. What elements in the domain of k o h have image 4?

Starting from the function y = x, describe how the function y = (5x – 3)2 is constructed.

Show how to deconstruct this function back to the original function.

Show that the function 2g( ) 1x x= − defined on the domain 0 ≤ x ≤ 1 is self-inverse.

Assessment


If 2f( ) 2, g( )x x xx

= + = are defined on the positive real numbers, find:

a. f–1(x)

b. g–1(x)

c. (g o f)(2)

d. (f o g)(1)

e. (f–1 o g–1)(2)

f. (g–1 o f–1)(5)

Unit 12F.6

257 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.7 | Measures © Education Institute 2005

GRADE 12F: Measures

Mensuration

About this unit This unit is the only one on measures for Grade 12 foundation. It provides an opportunity for revision of work on compound measures as well as extending students' experience to working with a frustum of a cone and with irregular areas.





Previous learning Students should be conversant with the formulae for lengths, areas and volumes associated with cone, cylinder and sphere. They should be proficient at working with compound measures.

Expectations By the end of the unit, students will identify and use connections between mathematical topics and break down complex problems into smaller tasks. They will calculate lengths, areas and volumes of geometrical shapes, using approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section. They will solve a range of problems involving compound measures, using appropriate units and dimensions. They will recognise when to use ICT and when not to, and use it efficiently.

Students who progress further will have strengthened their grasp of mensuration and prepared the way for calculus in the calculation of areas.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • computers with Internet access and spreadsheet software for students • calculators for students • cardboard with cutting and drawing tools

Key vocabulary and technical terms Students should understand, use and spell correctly: • cone, frustum, curved surface area

UNIT 12F.7 4 hours




Grade 11F standards CORE STANDARDS Grade 12F standards






11F.7.1 12F.9.1 Calculate lengths, areas and volumes of geometrical shapes.

2 hours

Mensuration and problem solving

2 hours

Irregular areas and prisms; problem solving with compound measures

Calculate lengths, areas and volumes of geometrical shapes. 12F.9.2 Use approximation methods to calculate the area of an irregular

two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section.

12AQ.8.712AS.10.6

11F.7.3 Work with SI units and compound measures including density, average speed and acceleration, measures of rate, and population density (number of people per unit area), using appropriate units and dimensions.

12F.9.3 Solve problems involving compound measures, using appropriate SI units and dimensions.

Use the trapezium rule to find an approximation to the area represented by the definite integral of a particular function when it is not easy or possible to integrate the function.

Unit 12F.7


Activities


2 hours

Mensuration and problem solving Identify and use interconnections between mathematical topics.

Calculate lengths, areas and volumes of geometrical shapes.

Solve problems involving compound measures, using appropriate SI units and dimensions.



The area and volume of a frustum of a cone

Discussion Ask students to remind other members of the class about the formulae already studied for areas and volumes. They should remember those studied in earlier grades: • area of the curved surface of a cylinder; • area of the curved surface of a sphere; • volume of a cylinder; • volume of a pyramid; • volume of a cone; • volume of a sphere.

Ask students how to calculate the surface area of the cone, using only dimensions of the cone. If there are no suggestions, pose the simpler question of how to calculate x (see the top two diagrams on the right). As a result of the discussion, derive the expression (see below) for the curved surface area of the cone. Ask students how to relate h, s and r; if necessary draw the triangle with h, s and r as sides of a plane figure.

2Area where 2

2c πs c πrπsπrs

= × =

=

Activity 1 Give students the chance to work on the new formula in context.

Activity 2 Show students a frustum of a cone (perhaps a cheap lampshade) and ask them to discuss how it is constructed geometrically. If a hint is necessary, open out the frustum after splitting the side. Also suggest that students: • sketch the cone that ‘completes’ it; • investigate how the dimensions of the two shapes are connected.

They may also need to be reminded of the formula for the area of the curved surface of a cone, and how the dimensions of a cone are interrelated by Pythagoras’ theorem.

On the web Formulae for the results in this section are quoted on a page of Mathworld at mathworld.wolfram.com/ConicalFrustum.html.


Unit 12F.7



Discussion

Continue the discussion with the case of a frustum of a cone. Ask for a method to calculate the volume of a frustum of specified upper and lower radii and height. If necessary, give a hint by drawing the missing part of the cone. Ask students how the dimensions of the cone (radius and height) can be specified; the radius is the base of the frustum, but it may take a moment for students to make the necessary connection with geometrical work. Suggest that similar triangles are the key to the height calculation.

( )

k bh k aka hb kb

k a b hbhbka b

=+

⇒ = +⇒ − =

⇒ =−

Now ask students how to complete the calculation (by subtracting the upper volume from that of the whole cone). As a follow-up question, ask how to calculate the frustum’s curved surface area.

Able students should be able to grasp both the algebra and the links with geometry. Less able students will be more content with the investigation and acquisition of the formula.

Activity 3 Set questions that explore the new formulae for a cone and that revise those learned in Grade 10F. Extend these to harder problem solving on the same theme. Include questions on flow rates relating to emptying tanks, etc.

Puzzles Examples of harder problems at this stage are: • If a full cylinder of water of height 15 cm and radius 4 cm is poured into an inverted cone of

base radius 7 cm and depth 20 cm, how deep is the frustum of empty space above the water in the cone?

• A bucket in the shape of a frustum of a cone of depth 40 cm, top radius 14 cm and bottom radius 10 cm contains water to a depth of 8 cm. How much does the water rise if a lead sphere of radius 4 cm is immersed in it?



Calculating irregular areas Remind students how to calculate the volume of a right prism.

Consider ways of calculating irregular areas drawn on a square grid. For example, • count grid squares included or partially included; • count only those included; • take the average; • multiply by the real area of a square.

Extend to working out the approximate volume of a right prism with an irregular cross section.

2 hours

Irregular areas and prisms; problem solving with compound measures Use approximation methods to calculate the area of an irregular two-dimensional flat surface and the volume of a prism with a constant, but irregular-shaped, cross-section.


Activity 4: Which is more densely populated, Doha or London or Barcelona?

To answer the question you will need to define the terms involved: for example, population

density is defined as population ,area

or number of people per square kilometre. Research figures

for population, and relate them to area.

For example, the website www.citypopulation.de/Qatar.html has some manageable data on the size of populations of the main places in Qatar. The area of each locality is also given in square kilometres. Students could use this real data to compare and contrast the population density (number of people per square kilometre) for various places in their country and for Qatar as a whole.

Use maps to obtain approximate figures for other areas corresponding to the figures of population. Attempt the calculation for one or more cities, dividing and sharing the work between students. To find the approximate area of a widespread city such as London, Sydney or Los Angeles will require a map with a square grid drawn on it.

There is a map with a zoom facility for Barcelona at www.bcn.es/guia/welcomea.htm. It shows all the districts of the city. Relevant population statistics (for other cities as well as Barcelona) are quoted on www.citypopulation.de/Spain.html.

A similar map with a zoom facility centring on Doha can be found at encarta.msn.com/map_701512116/Doha.html.

A street map for Doha with zoom facility is at uk.multimap.com/wi/337664.htm.

On the web Barcelona at www.bcn.es/guia/welcomea.htm

Doha at encarta.msn.com/map_701512116/Doha.html

Activity 5: Loss of water from the Dead Sea ‘In less than fifty years the level of the Dead Sea has declined from 395 metres below sea level to 410 below.’ The quotation is taken from an Internet article on the Peace Conduit, which discusses ways in which the level could be restored. • How can the loss of water be quantified?

You will require a map of the Dead Sea and an area calculation to work out that quantity.

On the web The article referred to is on www.mfa.gov.il/MFA/MFAArchive/2000_2009/ 2002/8/Israel%20and%20Jordan%20Launch%20Global%20Campaign%20to%20Save%20t

Activity 6: The trapezium rule The calculation of the area under a graph using the trapezium rule is an extension of this work. It is covered in depth in both Grades 12AQ and 12AS. It is also a good opportunity to demonstrate the value of a spreadsheet.

Able students especially will benefit from this pre-calculus work.


Assessment


On the pinboard, draw a trapezium that has a perimeter of 6 + 4√ 2.

This shape is designed using three semicircles.

The radii of the semicircles are 3a, 2a and a.

Find the area of each semicircle in terms of a and π, and show that the total area of the shape is 6πa2.

Find a when the area is 12 cm2, leaving your answer in terms of π.

A lampshade is in the form of a frustum of a right cone. The radius at the top of the shade is 10 cm and the radius at the bottom is 25 cm. Find the surface area of the material used for the lampshade.

Assessment


Research the area and population density of a large city.

Begin your work by calculating the approximate area by use of a suitable map.

Unit 12F.7

263 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.8 | Random variables and probability © Education Institute 2005

GRADE 12F: Random variables and probability

Probability models and calculations

About this unit This unit is the second of two on statistics and probability for Grade 12 foundation. It covers probability models, probability calculations and simulations.





Previous learning To meet the expectations of this unit, students should be able to calculate and use relative frequency and carry out and apply simple probability calculations.

Expectations By the end of the unit, students will solve routine and non-routine problems in a range of mathematical and other contexts, recognising connections between topics, and using mathematics to model and predict the outcomes of real-world applications. They will approach problems systematically, knowing when it is important to enumerate all outcomes. They will recognise when to use ICT, and do so efficiently. They will explain their reasoning orally and in writing. They will understand that a random variable has a range of values that cannot be predicted with certainty and will investigate common examples. They will measure the empirical probability (relative frequency) of obtaining a particular value of a random variable. They will use a simple mathematical model to calculate the theoretical probability of obtaining a particular outcome for a random variable associated with a set of events. They will calculate probabilities of single and combined events, and understand risk as the probability of the occurrence of an adverse event. They will use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

Students who progress further will have developed further fluency in all aspects of mathematical modelling.

Resources The main resources needed for this unit are: • overhead projector (OHP) • OHP calculator • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • computers with spreadsheet software for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • event, exclusive, independent, exhaustive • experiment, random variable, outcome • empirical probability, theoretical probability, conditional probability, relative

frequency, expected frequency, symmetry • simulation • risk

UNIT 12F.8 12 hours



12 hours SUPPORTING STANDARDS Grade 8, 9 and 11F standards







12F.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.


9.8.5 Use relative frequency as an estimate of probability and use this to compare outcomes of experiments.

12F.11.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

11F.8.4 Construct (relative frequency) histograms and plot cumulative frequency distributions, grouping continuous data when necessary.

12F.11.2 Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

9.8.7 Compare experimental and theoretical probability in different contexts.

12F.11.3 Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

8.8.9 12F.11.4 Understand risk as the probability of occurrence of an adverse event; investigate some instances of risk in everyday situations, including in insurance and in medical and genetic matters.

12AQ.10.4

2 hours

Empirical probability

1 hour

Mutually exclusive and exhaustive events

1 hour

Independent events

2 hours

Probability trees

2 hours

Simulation

4 hours

Risk Use problem conditions to calculate theoretical probabilities for possible outcomes. 12F.11.5 Understand when two events are mutually exclusive, and when a set

of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

Know that a probability distribution for a random variable assigns the probabilities of all the possible values of the variable and that these values total to 1; use a simple mathematical probability distribution to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

Unit 12F.8


12 hours SUPPORTING STANDARDS Grade 8, 9 and 11F standards



9.8.6 Know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B.

12F.11.6 Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.

12AQ.10.9

12F.11.7 Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

8.8.10 List systematically all the possible outcomes of an experiment.

12F.11.8 Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

Know that: • when two events A and B are mutually

exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone;

• two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B);

• when two events A and B are not mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where P(A) is the probability of event A alone, P(B) is the probability of event B alone and P(A ∩ B) is the probability of both A and B occurring together.

12F.13.1 Use coins, dice or random numbers to generate models of random data.

12AQ.10.11 Know that in general if event B is dependent on event A, then the probability of A and B both occurring is P(A ∩ B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given that A has occurred.

12F.14.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.


Activities


2 hours

Empirical probability Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

Probability and long-term relative frequency Ask students what they mean by stating that the probability of obtaining a head when tossing a coin is 1⁄2. From the discussion, draw out: • the common misunderstanding that if one toss is head the next is more likely to be tail; • the correct perception that any experimental justification requires a large number of tosses; • the idea that the symmetry of a coin justifies the claim.

To establish that a large number of tosses is needed in order to state with confidence that probability is equal to relative frequency may require an experiment to justify it. Simulations are a way to generate a lot of data, e.g. Coin tossing (nlvm.usu.edu/en/nav/vlibrary.html) or Adjustable spinner (illuminations.nctm.org/tools/index.aspx).

Then establish that there are two ways of modelling such a long-term relative frequency: it is necessary to do an experiment or to use symmetry.

An example of the first situation is the experiment of throwing drawing pins into the air and finding the proportion that land pin down. It is not possible to predict the outcome by considerations of symmetry; indeed, different types of drawing pin may lead to different results. However, the symmetry argument can be used with coins, playing cards, spinners, etc.

Finish the discussion (or round off the experimental session) by defining expected frequency as probability multiplied by number of experiments.

To fix ideas of probability if required, do an experiment with coins or drawing pins. Ask students to record in groups the proportion of coin tosses that produce heads.

If students work in groups, make sure that they use identical materials. Then bring together the results with a cumulative table like the one below.

No. of tosses 20 40 60 80 100

Proportion of heads

When plotted on a graph these values should (usually!) show that the proportion of heads settles on 0.5 approximately as the number of tosses progressively increases. This is an impression that students must acquire if they are to grasp the interpretation of probability. The resulting graph can easily be displayed on a graphics calculator.

The same approach can be taken if students are tossing drawing pins, with results recorded in a table showing how many land pin down.



Approach a problem systematically, recognising when it is important to enumerate all outcomes.

Exercises Set short exercises which bring out three issues: • calculation of probabilities by symmetry (such as for coins, playing cards, spinners); • interpretation of probability; • prediction of expected frequencies.

In doing this, see that students: • do not express probabilities as percentages; • use exact arithmetic when appropriate; • focus on the fact that probability necessarily lies between 0 and 1 and that the two extreme

values have interpretations in terms of impossibility and certainty.

On the web Math Goodies has a page on elementary probability theory (www.mathgoodies.com/ lessons/vol6/intro_probability.html).

Unit 12F.8



1 hour

Mutually exclusive and exhaustive events Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.


From intuition to precise language

Class discussion Introduce students to the ideas of exclusive and exhaustive events. These are ideas of which they probably already have an intuitive grasp, and the main task is to establish and use the vocabulary. Do this by using many examples; test students’ ability to recognise the right word to use in many different contexts.

Examples • Toss a coin and consider the events: – the coin is heads – the coin is tails (exclusive and exhaustive). • Shuffle a pack of playing cards, draw a card at random, and consider the events: – the card is black – the card is a face card (neither exclusive nor exhaustive). • Throw a dice and consider the events: – the number shown is even – the number shown is 5 (exclusive but not exhaustive). • Spin a spinner with the digits 1, 2 and 3, and consider the events: – the score is prime – the score is odd (exhaustive but not exclusive).

Additive probability law

Class discussion Round this part of the unit off by establishing the additive probability law: P(A ∪ B) = P(A) + P(B) when A and B are mutually exclusive

Associate this idea with a suitable probability space diagram (you may want to remind them of their work on sets and Venn diagrams in Grade 10).

Consider the outcomes when two non-biased, six-sided dice are rolled. Represent the probability space by a 6 by 6 grid, and consider the pair of events: • the first dice shows 4 • the first dice shows 6

These two do not overlap, so P(4 or 6) = P(4) + P(6).



By contrast, P(4 on the first or 6 on the second) ≠ P(4 on the first) + P(6 on the second).

The two events overlap in this case.

More able students may appreciate the fuller version of the additive law, easily justified – at least visually – by a Venn diagram of the probability space. More able students also will see the analogy between an event and a subset implicit in this.

The full version of the law, which holds in all cases, is P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

Puzzle A number of balls are placed in a bag. The balls are identical except for bearing the numbers 2 to 8; every ball has one number and no two have the same. One ball is drawn at random. • How many balls are there in the bag to start with? • What is the probability that the drawn ball has a 2 on it? • What is the probability that the number on the ball drawn is prime? • What is the probability that the number on the ball drawn is odd? • What is the connection between the three previous answers?

On the web Combining probabilities is a topic on Waldo’s Interactive Maths pages at www.waldomaths.com; follow the prompt from 14–16.

1 hour

Independent events Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

Precision without confusion

Class discussion Introduce the idea of independent events. Do this by focusing on successive events where the probability of the second is (or is not) affected by the first.

Examples • Removing balls from a bag with replacement and shaking each time leads to independent

events. • The same without replacement leads to dependent events.

Do these experiments with some actual balls in a bag (e.g. 9 red, 3 blue). Follow with more thought-provoking cases such as drawing a card from a pack and considering the events:

Independent Dependent

the card is black the card is black

the card is a face card the card is a spade

Then establish the multiplication law: P(A ∩ B) = P(A) × P(B) provided that the events A and B are independent.

Illustrate this with the case of the drawn playing card just considered.

Students may remark that exclusive events can be represented in a Venn diagram. The same is not true for independence, but the idea anticipates probability tree diagrams.

The playing card example of independence shows clearly how this law works, despite a feeling on the part of students that since the first event in that case does affect the possible outcomes for the second, the events may in fact not be independent.



Exercises Set questions that explore these ideas in simple terms. Include: • simple probability calculations that exploit the idea of independence, such as: – finding the probability of drawing a spade from a pack of cards and rolling a 6 on a dice; – finding the probability of obtaining ten heads in ten tosses of a coin; – finding the probability of guessing correctly a four-digit number (consisting of four digits,

including zero, in any order); • testing for independence, for example: – When rolling a dice do you think the two events (getting a prime number, getting a

multiple of 3) are independent? Why?

2 hours

Probability trees Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

Problem solving

Class discussion Set students a couple of questions from a standard exercise on probability trees, but without further assistance.

Example Feruk has ten coins in his pocket, six identical bronze ones and four identical silver ones. He draws two coins from his pocket. What is the probability that one is bronze and the other silver?

Able students may manage an answer to this but some may not. Of those who do, many will get 4⁄15 rather than the correct 8⁄15. This may well provoke argument. This motivates the introduction of a tree diagram to clarify the situation.

Introduce the tree diagram for the question, stressing the correct convention for labelling the branches with probability values and events. Demonstrate the multiplication rule for finding the probability associated with any particular path, and show the two paths associated with the question being considered.

Problems Set some problems that become progressively harder. Include: • situations in which two successive independent events occur (such as drawings with

replacement, successive sets of traffic lights, etc.); • situations in which two successive dependent events occur; • more complicated three-stage problems.

On the web GCSE Bitesize has a section on conditional probabilities at www.bbc.co.uk/schools/ gcsebitesize/maths/datahandlingh/ probabilityhrev1.shtml.

Examples of problems • A bag contains 5 red and 5 white discs. What is

the probability that the first disc taken out is red? The first disc is red, and is not replaced. How

many discs in the bag: (a) remain; (b) are red? What is the probability that a second disc taken

out is red? Draw a tree diagram of events. What is P(R, R)? • Mai buys 5 tickets in a raffle. 100 tickets are

sold, and there are 2 prizes. In the draw, the first ticket is not replaced. Make a diagram to show the possibilities.

Calculate the probability that Mai will win: (a) 2 prizes; (b) only 1 prize; (c) no prize.

• Cards are drawn at random, one at a time, from a pack of 52, and not replaced. Calculate the probability that: (a) the first card is an ace; (b) the first two cards are aces; (c) the first three cards are aces; (d) the first four cards are aces; (e) the first five cards are aces.



Class discussion Round off this section by introducing the conditional probability rule: P(A ∩ B) = P(A) × P(B | Α)

Point out that students have now grown familiar with it, as they used it unknowingly in trees. The only new point to make is to explain the ~ notation and that: P(B | Α) is the probability of B given that A has already occurred.

Challenge In a television quiz show, a contestant has to choose one of three doors. Two conceal booby prizes of no value, while the other conceals a large sum of money that the contestant hopes to win. Once the contestant has made a choice of door, but before that door is opened, the host opens one of the other two doors to reveal a booby prize, and then offers the contestant the chance to change their mind, that is to choose the remaining door instead of the first choice. Which should the contestant do?

If time permits, give the question to students to discuss. If you need to give a hint, ask students to work out the probability of winning on each of the two strategies separately, i.e. • to stick to the original choice; • to change. Whichever gives the greater probability is the correct answer.

Using ICT Try some of the interactive probability problems on www.cut-the-knot.org, e.g. the Lewis Carroll problem.

A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?

2 hours

Simulation Use coins, dice or random numbers to generate models of random data.

Use a calculator with statistical functions to aid analysis for large data sets, and ICT packages to present statistical tables and graphs.


Discussion Ask students about their expectations when tossing a coin. They may suggest: • it is impossible to predict the next toss; • there is a probability model of a fair coin; • to verify any probability value requires a large number of tosses.

Explain that models help in considering long-term trends (as moving averages do) but that the sort of variation which occurs in practice can be experienced only by doing experiments.

Explain that a simulation assumes that the underlying model is correct. It is useful to generate some data to represent the variation that would happen in practice.

Ask these questions: • In tossing 100 coins the number of heads is expected to be 50. In practice, how often would

you expect to get, say, 55? • Experience shows that the average queue in a doctor’s surgery is 6 patients. How often are

there 15 people waiting? What implications are there for the size of the waiting area?



Discuss the basis of simulation using random numbers by an example. Explain to students what a table of random numbers is, and agree a probability distribution for coin tossing (that the probability of a head is 0.5, and of a tail is 0.5, for example).

Explain how to allocate random digits to simulate those, by considering 0, 1, 2, 3 and 4 to be occurrences of head, and the rest of the digits occurrences of tails.

Get students to do a small-scale experiment using the random number keys of their calculators.

Explain that coins and dice can be used to simulate too.

Experiment Once the idea is clear, and also the implications in terms of data management, move on to using a computer to simulate the experiment. Alternatively, get students to work in teams to generate sufficient data. Require students to present their data using the techniques from earlier units and to summarise their conclusions.

In this sort of work, get students first to be clear about what questions they wish to ask, and to take account of those questions when they are designing their experiments.

Simulation using a spreadsheet such as Excel The extract from a spreadsheet in the notes on the right shows one simulation of a coin-tossing experiment. • The first column is the index number of the throw; there are 22 of these, but in practice a

much larger number should be used. • The second column simulates the throw of a coin: 0 denotes tail and 1 head. This is done by

B2=INT(0.5+RAND()), and replication down the column. • The first pivot table below shows an analysis of the results: there were 11 heads in this

experiment. • The third column counts the run length so far (the number of consecutive heads or tails).

This is done by entering 1 into C2 and C3=IF(B3=B2,C2+1,1) with replication down to C23. • The fourth column records the final run lengths by setting D2=IF(C3>C2,,C2) with replication

down to D22. The last cell D23 is blank, since without C24 it is impossible to tell whether the current run has finished. Ignore the zeros in column D.

• The second pivot table records the results. Sufficient data should provoke some interesting questions. Ask students to pose some, for example: • What is the distribution of run lengths? • What is the waiting time to (say) the fifth head?



4 hours

Risk Understand risk as the probability of occurrence of an adverse event; investigate some instances of risk in everyday situations, including in insurance and in medical and genetic matters.




Project work

Class discussion Discuss with students their understanding of the term risk (the probability of the occurrence of an adverse event). This may be applied to travel, for example, where students may have heard that air travel is the ‘safest’ mode. Ask students what such a statement may mean, and whether they consider air travel to be risky. Ask whether they think it is less risky to travel by road.

Try to establish some sort of agreed approach to the questions.

The website www.bast.de/htdocs/fachthemen/irtad/english/englisch.html is a useful source. This contains an overview of international road traffic and accident data and includes calculated risk values for the year 2002.

The World Health Organisation publishes an annual report with statistical annexes, which are offered for downloading as Excel files (see www.who.int/whr/en).

Choice of project Ask students to consider a particular area of risk of their choice. For example: • travel mode risks; • health risks – be specific; • accident risks in sport, such as sprains in tennis.

Ask students to research this and to provide a report.

Before starting, check that students have: • provided a clear question to research; • identified a data source to work on.

Stress to students the importance of sourcing, editing and summarising their data carefully, and of making their conclusions clear.

On the web There are statistics available on a variety of subjects (e.g. on the UK National Statistics site at www.statistics.gov.uk).

Note that for classroom use many sites offer only PDF files, so it is important to find sites that offer Excel files of data. An example of what is available from this site is www.statistics.gov.uk/STATBASE/ ssdataset.asp?vlnk=8397.

Also try the United Nations website (cyberschoolbus.un.org/infonation3/menu/ advanced.asp). This allows data from groupings of countries to be selected and compared (e.g. from North Africa and the Middle East).

More data sources are listed on the Oundle-TSM site at www.tsm-resources.com/mlink.html#stats.


Assessment


A set of 24 cards is numbered with the positive integers from 1 to 24. If the cards are shuffled and if only one is selected at random, what is the probability that the number on the card is divisible by 4 or 6?

A. 1⁄6 B. 5⁄24 C. 1⁄4 D. 1⁄3 E. 5⁄12

TIMSS Grade 12

One thousand people selected at random were questioned about smoking. The results of this survey are summarised in the table on the right.

Calculate the probability that a randomly selected respondent is male and smokes.

TIMSS Grade 12

Smokers Non-smokers

Males 320 530

Females 20 130

Two six-sided dice, each numbered from 1 to 6, are thrown and the total score on the two dice is found. Assuming that either dice is equally likely to show any of its six faces, what is the probability that the total score is greater than 4 and less than 10?

Assume that it is equally likely for a woman to give birth to a girl as it is to give birth to a boy. What is the probability that a woman with six children has four girls and two boys? What is the probability that if another woman has four children they are all boys? A woman with three daughters is going to have a fourth child. What is the probability that the fourth child will be a boy?

A warning system installation consists of two independent alarms having probabilities of operating in an emergency of 0.95 and 0.90 respectively.

Find the probability that at least one alarm operates in an emergency.

A. 0.995 B. 0.975 C. 0.95 D. 0.90 E. 0.855 TIMSS Grade 12

A company makes computer disks. It tested a random sample of disks from a large batch. The company calculated the probability of any disk being defective as 0.025. Naima buys two disks.

a. Calculate the probability that both disks are defective.

b. Calculate the probability that only one of the disks is defective.

c. The company found three defective disks in the sample they tested. How many disks were likely to have been tested?

Assessment


The probability of dying of cancer is 1⁄3. What is the probability that, if three people are chosen at random, two of them will die of cancer? What is the probability that none of them will die of cancer?

Unit 12F.8



A computer game has nine circles arranged in a square. The computer chooses circles at random and shades them black. At the start of the game, two circles are to be shaded black.

a. Show that the probability that both circles J and K will be shaded black is 1⁄36.

b. Halfway through the game, three circles are to be shaded black. Here is one example of the three circles shaded black in a straight line. Show that the probability that the three circles shaded black will be in a straight line is 8⁄84.

c. At the end of the game, four circles are to be shaded black. Here is one example of the four circles shaded black forming a square. What is the probability that the four circles shaded black form a square?

.

In a class of 35 students, the probability that a student picked at random is taller than 1.8 metres is 0.2 and the probability that the student wears spectacles is 0.3.

What is the probability when three students are chosen at random that two are over 1.8 metres in height and that one of them wears spectacles?

On a tropical island the probability of it raining on the first day of the rainy season is 2⁄3.

If it does not rain on the first day, the probability of it raining on the second day is 7⁄10.

If it rains on the first day, the probability of it raining more than 10 mm on the first day is 1⁄5.

If it rains on the second day but not on the first day, the probability of it raining more than 10 mm is 1⁄4.

You may find it helpful to fill in the tree diagram before answering the questions below.

a. What is the probability that it rains more than 10 mm on the second day, and does not rain on the first?

b. What is the probability that it has rained by the end of the second day of the rainy season?

c. Why is it not possible to work out the probability of rain on both days from the information given?

20 per cent of the population of a country has a particular disease. A test can be given to help determine whether people have the disease. The probability that the test is positive for those who have the disease is 0.7. But there is a 0.1 chance that a person who does not have the disease registers positive on the test.

a. Find the probability that an individual selected at random tests positive, but does not have the disease.

b. Another person is chosen at random. Calculate the probability that the test result for this person is positive.

Do an investigation using random numbers to simulate waiting times at a doctor’s surgery.



Transforming functions

About this unit This unit is the fourth of five on algebra in Grade 12 foundation. In the unit, students apply geometrical work on transformations to graphs, and learn how the same effects can be achieved by algebra.





Previous learning To meet the expectations of this unit, students should already be able to transform elementary geometrical figures by using sketches and by using ICT. They should be confident at simple algebraic manipulations.

Expectations By the end of the unit, students will identify and use connections between mathematical topics. They will apply combinations of transformations to the graph of the function y = f(x).

Students who progress further will quickly and confidently make connections between geometric and algebraic transformations.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • graph plotting software such as:


• computers for students with Internet access and graph plotting software • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • reflection, stretch, translation • algebraic and geometric transformation

UNIT 12F.9 6 hours



6 hours SUPPORTING STANDARDS Grade 9 and 12F standards




9.5.6 Identify a single transformation mapping a 2-D shape onto its image: reflection, rotation, translation or enlargement by a positive integer scale factor; find a line of reflection, centre or angle of rotation, scale factor or centre of enlargement in simple cases.

12F.5.7

9.5.7 Identify and draw, on paper and using ICT, the enlargement of a simple plane figure by a positive fractional scale factor; identify the scale factor as the ratio of two corresponding line segments.

Understand the transformations of the function y = f(x) given by: • y = f(x) + a, representing a translation by a in the positive

y-direction; • y = f(x – a), representing a translation by a in the positive

x-direction; • y = af(x), representing a stretch with scale factor a parallel

to the y-axis; • y = f(ax), representing a stretch with scale factor 1/a parallel

to the x-axis;

use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

6 hours

Transformations of graphs

9.5.8 Use ICT to explore transformations. 12F.3.1 Develop further confidence in all the

calculation skills established in Grades 10 and 11.

Unit 12F.9


Activities


Transformations

Discussion Ask students what they remember about transformations. Use the discussion to review: • translation; • reflection; • rotation; • stretching (shear); • enlargement. Except for rotation and enlargement, students will study these again in what follows.

To make sense of what follows, students need to be able to sketch the image of transformed diagrams or graphs with ease.

Stretching may be new to students since it does not figure in the Grade 9 standards on transformations.


Use Autograph and a whiteboard to sharpen student responses. For example, produce a figure and ask students to mark its new position using a suitable marking system. Do this as much as is necessary to establish the ideas with confidence.

Exercises As this work has not been covered since Grade 9, it may be necessary to give students practice to bolster confidence in these skills.

Refer to the Grade 9 units for suitable material.

Use of Autograph to illustrate reflection in the y-axis and a stretch of factor 0.5 parallel to the x-axis

Practice Remind students how to use Autograph, by graphing y = 3x and y = –2x + 1.

Give students a range of functions to graph using graph plotting software (better) or graphics calculators (acceptable). For examples, see the notes on the right.

For trigonometry examples, use radian measure (to avoid large numbers).

Examples for students to graph • y = x2 • y = 5 – 4x – x2 • y = 2x3 – 5x2 + 4x – 7 • y = sin x • y = tan (2x – 3)

6 hours Transformations of graphs


Understand the transformations of the function y = f(x) given by: • y = f(x) + a, representing a

translation by a in the positive y-direction;

• y = f(x – a), representing a translation by a in the positive x-direction;

• y = af(x), representing a stretch with scale factor a parallel to the y-axis;

• y = f(ax), representing a stretch with scale factor 1/a parallel to the x-axis;

use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

Investigations Ask students to investigate systematically what happens geometrically in the case of each of the transformations on the right. Do an example to illustrate how to effect the transformation on the calculator or graph plotting software. Give students advice on how to record the work as it proceeds. This is to avoid them being so overwhelmed by the quantity that they are unable to see resulting patterns. If using graph plotting software, save the files; with a calculator, make indexed sketches.

Give this the time it needs. Encourage discussion before summarising the conclusions.

Transformations to investigate • Change x to (x + 3). • Change x to (x – 2). • Change f(x) to f(x) + 3. • Change f(x) to f(x) – 2. • Change x to (3x). • Change x to (–2x).

• Change x to (1⁄2 x). • Change f(x) to 3 f(x). • Change f(x) to –2 f(x). • Change f(x) to 1⁄3f(x).

Unit 12F.9



Autograph is well adapted to this work. The diagram on the right shows the graphs:

y = 5 – 4x – x2 y = 5 – 4(x – 2) – (x – 2)2

Class discussion Students met the idea of composition (and decomposition) in Unit 12F.6, on composite and inverse functions. The aim now is to connect this idea with the transformations just considered. Discuss an example and spell out the steps of the decomposition in parallel with the geometric development.

Example y = 2 – 4 sin (5x + 3)

x → sin x start with the standard sine wave (using radian measure)

x → sin (x + 3) translation 30

−⎛ ⎞⎜ ⎟⎝ ⎠

x → sin (5x + 3) stretch parallel to Ox with scale factor 1⁄5

x → –4 sin (5x + 3) stretch parallel to Oy with scale factor –4

x → 2 – 4 sin (5x + 3) translation 02⎛ ⎞⎜ ⎟⎝ ⎠

The diagrams on the right show the various stages of the process. Tell students that they should be able to sketch the steps of this by careful argument and knowledge of the transformations.

y = sin x

y = sin (x + 3)

y = sin (5x + 3)

y = –4 sin (5x + 3)

y = 2 – 4 sin (5x + 3)



Exercises Set questions that require this technique. Begin with explicit equations, so that the result (final or step-by-step) can be checked on a graphics calculator.

Increase the difficulty by providing sketches of unspecified functions so that students must think geometrically. To be successful students must be able to do this without recourse to squared paper.

Example 1 Provide a sketch of an unknown function y = f(x) and use Autograph to sketch: • y = f(2x) • y = 3f(x + 3)

Example 2 Use graphics calculators to explore possible equations for these graphs.


Assessment


Curve A is the graph with equation y = x2. On the same axes, sketch the graph with equation y = 2x2. Curve A

Curve B is the translation, one unit up the y-axis, of y = x2. What is the equation of curve B?

Translate curve B two units to the left. What is the equation of this new curve?

Curve B

A function is defined by f(x) = x2. Describe the functions:

a. f(x – 2)

b. f(x + 1)

State how the graphs of each function relate to the graph of y = f(x). Give the defining equation for each function.

Transform the curve y = x3 into the curve y = 5x3. Describe the effect of the transformation.

The curve is then translated one unit in the positive x-direction. What is the equation of this new curve?

Assessment


Describe in words how the graph of y = 1/x is transformed into the graph y = 4 + 5/x.

Sketch each graph on the same set of axes.

Explain the difference between:

a. the functions y = cos x° and y = cos (x + 45)°

and:

b. the functions y = cos x° and y = 2 cos x°.

Unit 12F.9

281 | Qatar mathematics scheme of work | Grade 12 foundation | Unit 12F.10 | Using vectors © Education Institute 2005

GRADE 12F: Using vectors

Vectors, including scalar product

About this unit This unit, the only one on vectors for Grade 12 foundation, is an introduction to elementary vector methods.





Previous learning To meet the expectations of this unit, students should already be able to use coordinates confidently, and use trigonometry and Pythagoras' theorem to solve triangles.

Expectations By the end of the unit, students will identify and use connections between mathematical topics. They will aim to generalise. They will begin to use vectors to solve physical problems.

Students who progress further will use vectors as a conceptual tool that extends ideas familiar in one dimension to two and three.

Resources The main resources needed for this unit are: • overhead projector (OHP) and transparencies • Internet access, computer and data projector • (optional) dynamic geometry system (DGS) such as:


• (optional) computers with Internet access and dynamic geometry software for students

• scientific calculators for students • squared paper for coordinate grids

Key vocabulary and technical terms Students should understand, use and spell correctly: • vector, scalar, component, magnitude, direction, sense • vector sum, vector triangle, resultant • section formula, position vector, scalar product • collinear

UNIT 12F.10 7 hours





EXTENSION STANDARDS Grade 12AS standards



12F.8.1 Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

4 hours

Vector algebra

3 hours

Scalar products 12F.8.2 Interpret a translation as a vector displacement; know that a vector

displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

12F.8.3 Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

12F.8.4 Multiply a vector by a scalar and know that this amounts to stretching the vector; calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.

11F.6.2 Solve right-angled triangles using the standard trigonometric ratios, including tan θ = sin θ /cos θ, and/or Pythagoras’ theorem.

11F.6.3 Know and use the sine rule and the cosine rule to solve triangles.

12F.8.5 Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

12AS.13.8

12F.8.6 Solve physical problems using vectors.

Find the vector equation of a straight line in the form r = a + λb, where r is the position vector of any point on the line, a is the position vector of a given point on the line, b is a vector in the direction of the line and λ is a variable scalar.

Unit 12F.10


Activities


Introducing vectors

Activity 1: to familiarise students with the idea of a free vector Equip students with coordinate grids and sets of transparencies, with vectors represented as scaled arrows with pairs of components written on them. Get students to: • place the arrows on the coordinate grid correctly oriented; • investigate the relationship between length and components.

In this and all other activities in this unit, see that students are equally at home with reading

column form (e.g. 23

⎛ ⎞⎜ ⎟−⎝ ⎠

) and using unit vectors i and j (e.g. 2i – 3j). Encourage students

always to use the column form in writing, since it is so much easier to read and interpret.

When appropriate, extend the work to the three-dimensional equivalent forms.

The nose-to-tail arrangement for adding two vectors


Activity 2: to familiarise students with vector components Have students play a blind treasure hunt. Provide them with a map on a coordinate grid. On the map is a collection of obstacles and pathways (such as the buildings and roads of a housing area). They must use vectors to direct each other around the possible clear routes to move from one place to another.

The two vectors marked a must have the same length and direction, so the quadrilateral is a parallelogram. So the other two sides represent the same vector b. Then the diagonal is either a + b or b + a, so they are equal vectors too. This is the commutative law.

Activity 3: to familiarise students with the process underlying vector addition

Draw a figure (such as an elephant) using an outline of straight line segments on a coordinate grid so that each line joins two grid points. Label the points with capital letters (A, B, C, etc.),

and list the vectors that represent the line elements in the form, for example, 2AB 3⎛ ⎞= ⎜ ⎟⎝ ⎠

uuur. Then

get students to draw the figure from the list of the line segments and components.

4 hours

Vector algebra Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

Interpret a translation as a vector displacement; know that a vector displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

Calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.


Solve physical problems using vectors.

Multiply a vector by a scalar and know that this amounts to stretching the vector.

Activity 4: to consolidate addition and subtraction of vectors Give students more transparencies. Ask them to arrange the transparencies so that they illustrate sums of vectors and differences.

Challenge More able students who can master this easily can consider proving the commutative and associative addition rules by connecting this work with pure geometry. They will already appreciate that the work can generalise to three dimensions. These results are considered in Unit 11A.12.

Unit 12F.10



On the web To introduce the idea of magnitude and direction, see Car storm chaser and Airplane storm chaser (illuminations.nctm.org/tools/index.aspx)

Activity 5: using DGS Get students to use a dynamic geometry system (DGS) to demonstrate the effect of changing magnitude and direction in simulations of physical contexts.

There are examples of this kind of activity available on the web.

Vector notation Introduce the idea of position vector and the notation associated with it: • use capital letters for points of a diagram, including the origin O; • use small bold italic letters to denote the position vectors of such points, OA=a

uuur (when

handwriting vectors, underline the letters).

Stress that a number pair such as (4, 7) always represents a pair of coordinates and hence a

point, whereas a number pair such as 47⎛ ⎞⎜ ⎟⎝ ⎠

always represents a vector. The connection

between the two is that a point always has the column with the same number pair as its position vector.

Generalise these ideas to three dimensions.

There are several useful tools for vector problems; discuss these with students before starting exercises.

• AB = −b auuur

for any two points A and B.

• The section formula: if P divides AB in the ratio m : n, then we have ;n mm n

+=+

a bp and this

includes the case of external division where one of m and n is negative. • Three points A, B and C are collinear if and only if AB BCλ=

uuur uuur for some scalar λ, and this

applies for A, B, C in any order.

For less able students these results will be a challenge in two dimensions. Students that are more able will realise that the derivation of the section formula applies equally well in three.

Activity 6

Get students to match representations of different but parallel vectors (on drawings or

transparencies) to illustrate relations such as 3 124 2 8⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

.

Include cases that can illustrate the distributive law.

These results make good connections between pure geometry and vector methods. Able students especially should experience them.

Proof of the section formula can be found in Unit 11A.12, Vectors.

On the web A survey of elementary vectors that extends the work a little further than this unit is at www.physics.uoguelph.ca/tutorials/vectors/ vectors.html.

Some interactive Sketchpad resources are at mathforum.org/~klotz/Vectors/vectors.html.

Physical uses of vectors are discussed in the Physics Classroom at www.physicsclassroom.com/Class/vectors/ vectoc.html.



Challenge The proof of the distributive law given on the right can be offered to more able students. Again, it makes a connection with pure geometry.

Magnitude of a vector

Students will have easily mastered the idea that a vector in the form ab⎛ ⎞⎜ ⎟⎝ ⎠

has magnitude

2 2a b+ by using Pythagoras’ theorem. Show students (or challenge them to show) how this

distance formula generalises to three dimensions as 2 2 2 .ab a b cc

⎛ ⎞⎜ ⎟ = + +⎜ ⎟⎝ ⎠

Get students familiar

with this idea in preparation for the work on scalar product.

Problems Set questions that explore the ideas and the notation.

Examples • If A is the point (4, –5) and B (3, 10), find: – the position vector of A; – the vector which joins A to B; – the length of AB. • If P is the point (7, 9) and Q (–4, 3), find: – p; – PQ;

uuur

– QP;uuur

– p – q; – q – p; – QP; – PQ.

Which of these are equal? Illustrate them in diagrams.

• The forces 432

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

newtons and 572

⎛ ⎞⎜ ⎟−⎜ ⎟−⎝ ⎠

newtons act on a particle.

Find the resultant and its magnitude. • ABCD is a parallelogram. M is the mid-point of AB, and T divides DM in the ratio 2 : 1. If

AD=uuur

u and AB=uuur

v , find the vector represented by ATuuur

in terms of u and v. Deduce that A, T and C are collinear, and find AT : TC.

The parallels mean that corresponding angles are equal, so OAB is similar to OA′B′. The common ratio of sides then gives:

OA OAk k′ ′= ⇒ =a auuuur uuur

and similarly k′ =b b

A B AB( )kk

′ ′ == −b a

uuuur uuur

But

A Bk k

′ ′ ′ ′= −= −

b ab a

uuuur

so finally

A B ( )k k′ ′ = − = −b a b kauuuur

Inclusion of examples in physical contexts are important and should be used to exploit students’ knowledge of other subjects.



3 hours

Scalar products Identify and use interconnections between mathematical topics.

Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

Multiplying vectors

Class discussion Students may already have raised the question of multiplying vectors; if so, start from that cue. Otherwise, start from the distributive law, where vectors and scalars multiply to give vectors. Show how to define the scalar product: a.b = | a | | b | cos θ

geometrically. Do a few examples to stress the following. • Point out the distinction in notation between a the vector and | a | its magnitude. • The scalar product is commutative, i.e. a.b = b.a. • If θ = 0, the scalar product is just | a | | b |. • If θ = π⁄2, the scalar product is just 0. • If θ is obtuse, the scalar product is negative.

On the web

Some interactive Sketchpad resources that cover scalar product are available at mathforum.org/~klotz/Vectors/vectors.html.

Continue to derive the more useful expression for scalar product in components.

In the diagram, let A be (a1, a2, a3) and B (b1, b2, b3). Then we have:

( ) ( ) ( )

1 1

2 2

3 3

2 2 2 21 1 2 2 3 3

2 2 2 2 2 21 1 1 1 2 2 2 2 3 3 3 3

AB

AB

2 2 2

b ab ab a

b a b a b a

b b a a b b a a b b a a

= −−⎛ ⎞

⎜ ⎟= −⎜ ⎟−⎝ ⎠

⇒ = − + − + −

= − + + − + + − +

b auuur

uuur

Now make the connection with trigonometry to bring in the cosine rule: 2 2 2

2 2 2 2 2 21 2 3 1 2 3

AB OA OB 2 OA OB cos

2

θ

a a a b b b

= + −

= + + + + + − a.b

uuur uuur uuur uuur uuur

Now equate the two expressions for 2

ABuuur

to obtain finally:

1 1 2 2 3 3a b a b a b= + +a.b

Continue to derive the result to use for calculation of the angle between two vectors:

1 1 2 2 3 3

1 1 2 2 3 3

cos

cos

θ a b a b a ba b a b a bθ

= = + +

+ +⇒ =

a.b a b

a b

Show this in use with an example.

The formula is fully derived in Unit 11A.12.

The formal derivation of the scalar product formulae is essential work for able students. Less able students will appreciate a more intuitive approach based on interactive resources, such as those listed above.



Practice Set exercises that: • drill the calculation of angles; • use the scalar product to establish the existence of right angles.

Examples • Use the scalar product to find ∠ABC, where A is (5, 4, 3), B (9, 3, –2), C (4, –1, 2). • Triangle PQR has vertices P (5, 7, –5), Q (4, 7, –3), R (2, 7, –4). Use the scalar product to

show that the triangle is right-angled. Can you do this any other way? • Given that the angle between vectors 2i – j + 3k and i + 3j – pk is 1⁄3 π, find p.

Extension Learning about properties of vectors and vector sums using dynamic software (see standards.nctm.org/document/eexamples/chap7/7.1/index.htm)


Assessment


A particle is at the point (6, 2). What is its position vector in terms of the unit vectors i and j in the x- and y-directions respectively? Calculate the length (magnitude) of this vector.

Four vectors a, b, c and d are given by a = 2i – 3j, b = 5j + k, c = 4i – 7k and d = 3i + j. Find a + b, b – c, a – b – c. Draw vector diagrams to represent a + d and a – d. What are the components of these two vectors in the i and j directions?

Find the magnitude of each of the vectors a and d. Calculate the angles between these vectors.

A particle moves with constant velocity from A to B. Its position vector at A is a = i + j and its position vector at B is b = 5i – 7j. Calculate the vector displacement from A to B.

If distance is measured in metres, show that the distance from A to B is 4√5 metres.

The particle takes 2 seconds to move from A to B. What is its velocity?

Assessment

Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. A particle of mass m kilograms is moving with constant acceleration a, measured in metres per

second per second. The total external force F acting on the particle is measured in newtons, and is the vector sum of the individual forces acting on the particle. The relationship between F and a is given by Newton’s second law of motion and is F = ma.

A particle of mass 2 kg is acted upon by two forces F1 = i – j and F2 = 3j. Find the acceleration of the particle and give its magnitude.

Two vectors a and b (a, b ≠ 0) are related by | a + b | = | a – b |.

What is the measure of the angle between a and b?

TIMSS Grade 12

In the figure, OACB is a parallelogram, and M is the mid-point of BC.

Vector OA = auuur

and vector OB = buuur

.

a. Find, in terms of a and b,

(i) BC;uuur

(ii) BM;uuur

(iii) OM.uuuur

b. Given that OX is a straight line and that OB = BX, prove that AMX is a straight line.

MEI

Unit 12F.10



Exponential and logarithm functions

About this unit This unit is the last of five on algebra in Grade 12 foundation. The unit offers work with exponential and logarithmic functions.





Previous learning To meet the expectations of this unit, students should already be able to measure gradients on a smooth curve and understand repeated compounding of interest.

Expectations By the end of the unit, students will use mathematics to model and predict the outcomes of real-world applications. They will develop and explain chains of logical reasoning, using correct mathematical notation and terms. They will generate mathematical proofs and identify exceptional cases. They will aim to generalise. They will use physical contexts to plot and interpret the graphs of exponential and logarithm functions.

Students who progress further will have met some key concepts for further work and will be motivated to study calculus at a later stage.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • graph plotting software such as:


• computers with spreadsheet and graph plotting software for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • logarithm, exponential • gradient • base of natural logarithms

UNIT 12F.11 10 hours







12F.1.6 Develop longer chains of logical reasoning, using correct mathematical notation and terms.


12F.1.8 Generate simple mathematical proofs, and identify exceptional cases.


11F.3.4 Investigate the problem of compounding interest more and more, and note that this tends to a limiting value; use this context to learn about the number e.

12F.5.8 Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.

12AQ.6.2 Solve exponential and logarithmic equations of the form ekx = A, where A is a positive constant, and ln kx = B, where B is constant.

10 hours Logarithms and exponential functions

11F.5.3 Draw the tangent line at a point on the graph of a function, calculate the slope of this line and interpret the behaviour of the function at that point, knowing whether the function is increasing of decreasing at the point, or stationary.

Unit 12F.11


Activities


Graphs and their gradients

Discussion and investigation Ask students to draw the graph of y = 2x. They will need to use a table of values and may need some assistance with scales to do this satisfactorily.

Once they have achieved that, draw the graph using Autograph or other graph plotting software.

Ask students what they expect to be different about the graph y = 3x. Draw that in Autograph too. Discuss the way the base (i.e. 2 or 3) affects the resulting graph. Note that the intercept on the y-axis is 1 in each case, and that upper and lower swap over as x goes from positive to negative.

Now use Autograph to draw the graph y = ax. By default Autograph will assign a default value of a = 1. Vary the constant value in Autograph (View → Constant Controller) so that students can see the variation in shape dynamically as the value of a increases or decreases, and that it coincides with theirs when a = 2.

Practice

Set questions on these ideas.

Examples • A man invests QR 1000 at 5% compound interest per year. Show that after two years his

investment is worth approximately QR 1102, and find an expression for his investment QR X after n years. Use your calculator to show the graph of that relation, and comment on its general features.

• A population of rabbits is modelled by P = 600 × 1.03m, where P is the size of population (i.e. the number of rabbits) and m is the time that has elapsed in months. Find the initial population’s size, and the size of the population after two and after three months. Use a graph of the function on your calculator to estimate how long the population takes to triple in size. Do you think such a model is realistic?

• A radioactive substance decays following the model N = 500e–0.001t, where N kg is the mass and t the time in days. Use your calculator to show a graph of this relationship.

Autograph display showing use of the constant controller to find an approximation for e (referred to later in the unit)

The trace function can be used to answer the first example.

The rabbit model shown in Autograph


10 hours

Logarithms and exponential functions Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.


Develop longer chains of logical reasoning, using correct mathematical notation and terms.


[continued]

• Use your graphics calculator to obtain a graph of y = 1.5x. Experiment with values of a to obtain the graph of y = ax for which the new graph is a reflection of the old in the line x = 0. Repeat this for y = 1.6x. Make a table of the values a and b for which y = ax is a reflection of y = bx. What is the general relationship between a and b? Can you prove that that general relationship holds?

1( ) 1

11

x x

x x

x

a ba b

abab

ab

−=⇒ × =⇒ =⇒ =

⇒ =

Unit 12F.11



[continued]

Generate simple mathematical proofs, and identify exceptional cases.


Using gradients

Discussion Students have met gradients of curves in Unit 11F.6; this section will build on that knowledge.

On the same axes in Autograph, show the graph y = ax and its gradient function. Ask students to explain how to interpret the picture. Make the connection between gradient and rate of increase; remind students of the connections between distance, speed and time on graphs to reinforce the idea.

On the web

MathsNet has pages on this and later sections: www.mathsnet.net/asa2/2004/c3.html#3.

There is also a page devoted to the properties of this and other exponential functions at Visual Calculus: archives.math.utk.edu/visual.calculus/ 0/exp_log.5/.

Now pose the question of how to get the two curves to coincide. It is easy to get the value 2.7 to achieve this. Tell students that a more precise value is 2.7183, and, if appropriate, mention also

that the number can be shown to be 1 11 1 ... e.2! 3!

+ + + + = They may have met this before in

work on repeated compounding of interest.

Show also that with the same value e the function y = Aex has the same property (i.e. curve and gradient curve coincide).

The Autograph display shown earlier illustrates the process of finding e by matching the curve and its gradient.

The gradient function key is located on the toolbars, showing a parabola with a tangent line.

Investigation • Use your calculator to show a graphs of y = 2x and y = e0.7x. What do you notice? • Find by experiment a value of k such that y = 3x and y = ekx coincide. You may need to

experiment also with the range and domain of your display to find a good value. • Make a table of pairs such as (2, 0.7) and (3, a) (where a is the value just found). • Plot the points represented by those pairs on a graph. • Can you derive the general relation between c and k if y = cx is identical to y = ekx?

The diagram shows the close correspondence between y = 2x and y = e0.7x.

Students should discover that any curve y = cx can be mimicked by a curve y = ekx for a suitable choice of k, and that c = ek.



From exponentials to logarithms

Class discussion Remind students of the relation between the graph of a function and that of its inverse: one is a reflection of the other in the line y = x. Use this idea to introduce the idea of the logarithm function; it is the inverse of y = ex. Focus on this by using the notation: y = ex ⇔ x = ln y

Point out that values of both functions are available on calculators, and that the work done so far shows that they suffice to handle all functions of the form ax.

Now develop the properties of this new function, using this notation to link it directly to the exponential function. a = ln p and b = ln q ⇒ p = ea and q = eb ⇒ pq = ea eb = ea+b ⇒ ln (pq) = a + b

= ln p + ln q

Graphs of the exponential function

and its inverse

Derive expressions for ln p

q⎛ ⎞⎜ ⎟⎝ ⎠

similarly. Use the special case of q = 1 to show that ln 1 = 0.

Ask students to simplify ln (xn) and ln (x–n), where n is a positive integer. Check that students begin by changing them to exponential functions. Discuss this, and conclude that: ln (xn) = n ln x and ln (x–n) = –n ln x

Invite students to conjecture how ln (xm/n) simplifies. When the correct answer is in view, allow sufficient discussion to show that this does not follow just by changing in the previous result n to m/n (because m/n is not a positive integer). Then demonstrate how to use the previous result to show that the equivalent result is nonetheless valid:

( )/

/

ln( )

ln ( )

ln( )ln

ln

m n

m n n

m

p x

np x

xm x

mp xn

=

⇒ =

==

⎛ ⎞⇒ = ⎜ ⎟⎝ ⎠

Practice Set questions that apply simple properties of logarithms, such as: • Express ln 6, ln 12 and ln 1.5 in terms of ln 2 and ln 3.

• Use logarithms to find k as a function of c if y = cx is identical to y = ekx. The second problem was considered earlier.



Logarithms to any base

Investigation On the calculator the function keys ln x and log x are available. Investigate the relationship between the values produced for different arguments x.

Graphics calculator displays of the use of log and ln keys

Class discussion Ask students for what values of x they can give a precise value for ln x (e.g. x = 1, e and other values such as e2). Introduce the problem of calculating (say) log3

2. To do this, express the problem in index terms and then take logarithms to a base available on a calculator (base e or 10). Thus

3

3

log 2

2 3ln2 ln3

ln2ln3ln2i.e. log 2ln3

p

p

p

p

=

⇒ =⇒ =

⇒ =

=

This is a special case of the general result loglog ,log

cb

c

xxb

= which is proved similarly and is a

good challenge for more able students.

With students, convert a value of their choice from base e to base 10.

The theory studied here does not answer the question of how the logarithms given on a calculator are worked out. Encourage students who persist with that question to consider the series

2 3

ln(1 ) ... ( 1 1)2! 3!x xx x x+ = − + − − < ≤

as a starting point.



Practice Set questions that cover all these ideas. See that as a result students are familiar with the use of the ln, log and ex keys on their calculators.

Examples • Calculate log4

5. • Solve the equation 3x = 4. • Solve the equation ln x + ln (x + 3) = ln 4. • Solve the equation 4x – 3 × 2x + 2 = 0. • Sketch on the same axes the graphs of y = ln x, y = ln (x + k) and y = ln (kx), making clear

how they are related.

• Find the image of the curve y = ex:

– after a translation parallel to the x-axis of ;0k−⎛ ⎞

⎜ ⎟⎝ ⎠

– after an additional stretch parallel to the y-axis of scale factor b. What happens when b = e–k?

Investigations Set questions that explore the use of the new function, including: • sketch graphs, using the graphics calculator; • scientific examples of growth and decay; • development of different models for comparison, e.g. a growth function with different

proposed parameters compared with linear variation – this is an opportunity for cross-curricular work.


Assessment


Given logb 2 = 1⁄3, logb

32 is equal to:

A. 2 B. 5 C. 35

− D. 53

E. 2

3log 32

TIMSS Grade 12

A radioactive element decomposes according to the formula y = yoe−kt, where y is the mass of the element remaining after t days and y0 is the value of y for t = 0.

Find the value of the constant k for an element whose half-life (i.e. time to decompose half of the material) is 4 days. A. 1⁄4 loge

2 B. loge 1⁄2 C. log2

e D. (loge 2)1/4 E. 2e4

TIMSS Grade 12

Assessment


The growth of the Internet since 1990 has been modelled by the function N = 0.2(1.8)t, where N is the number of users, counted in millions, t years from 1990.

Plot the graph of this function.

How many Internet users does the model predict for the year 2006?

When living organisms die, the amount of carbon-14 present in the dead matter decays exponentially according to the formula N = N0e–0.000121t, where N0 is the initial quantity and t is the time in years.

A bone uncovered at an archaeological site has 35% of its original carbon-14. Estimate the age of the bone.

After how many more years will the bone have only 25% of its carbon-14?

The number of bacteria in a colony of bacteria grows exponentially. At 1300 hours yesterday the number of bacteria was 1000 and at 1500 hours it was 4000. How many bacteria were there at 1800 hours yesterday? How many bacteria will there be at 1000 hours today?

TIMSS Grade 12

The Global Report estimated the population of the world in 1975 as 4.1 billion people and that it was growing at the rate of 2% per year.

Set up an equation to predict the world population t years from 1975.

Use this model to predict the world’s population in 2020. Discuss any assumptions you have made.

Earthquakes produce oscillations in the ground. The strength, S, of the quake is measured on the Richter scale and is given by S = log A, where A is the amplitude of the oscillation measured in millimetres on a calibrated seismograph.

What amplitude of oscillation corresponds to a major earthquake with a Richter scale value of 7.8?

What is the Richter scale value of an earthquake with an oscillation that has an amplitude of 2000 mm?

Unit 12F.11

mathematics units grade 12 foundation - csomathscience · pdf fileuse of graphics calculator...

Documents