Mathematics units Grade 10 advanced - csomathscience · PDF fileMathematics units Grade 10 advanced Contents 10A.1 Number 33 10A.7 Geometry 2 87 10A.2 Geometry 1 41 10A.8 Number and

Mathematics units Grade 10 advanced

Contents

10A.1 Number 33 10A.7 Geometry 2 87

10A.2 Geometry 1 41 10A.8 Number and algebra 1 97

10A.3 Algebra 1 49 10A.9 Statistics 2 105

10A.4 Statistics 1 59 10A.10 Algebra 3 111

10A.5 Geometry and measures 1 69 10A.11 Geometry and measures 2 121

10A.6 Algebra 2 77 10A.12 Number and algebra 2 129

Mathematics units: Grade 10 advanced 135 teaching hours

UNIT 10A.9: Statistics 2Data collection and analysisUse of secondary dataMeasures of central tendency andspreadScatter diagrams, line of best fit,positive and negative correlationUse of calculator; use of ICT statisticalpackages and databases10 hours

15% 30%

1st semester70 hours

2nd semester65 hours

UNIT 10A.2: Geometry 1Angle, shape and geometric reasoningInterior and exterior angles of polygonsCongruence and similarityProof10 hours

UNIT 10A.5: Geometry andmeasures 1BearingsPerimeter, area and volumeRadian measure; sectors andarcsSI units, rates and compoundmeasures10 hours

UNIT 10A.4: Statistics 1Planning surveysTypes of data, sampling, use ofprimary dataMeasures of central tendencyStem-and-leaf diagrams; box plotsHistograms for grouped continuousdata10 hours

UNIT 10A.7: Geometry 2Circle theoremsConstructionsLoci10 hours

UNIT 10A.11: Geometry andmeasures 2Pythagoras' theorem, includingproofTrigonometry, solution oftrianglesEquation of a circle10 hours

Reasoning and problem

solving should be integrated into each unit

UNIT 10A.0: Grade 9 revision3 hours

UNIT 10A.1: NumberPercentages, ratio, proportionalreasoningReal numbers; powers and roots; surdsStandard form; calculator use9 hours

UNIT 10A.8: Number and algebra 1Sequences, including use of graphicscalculator and spreadsheetSimple growth patterns; Pascal'striangleArithmetic sequences; sum of first nintegersGeometric sequences; recurringdecimals12 hours

UNIT 10A.3: Algebra 1Equations, identities, formulaeGeneralising rules of arithmeticSimplifying numeric and algebraic fractions,including denominators with surdsRearranging formulae; substitution offormulae into an expressionSolution of any linear equation, one unknown14 hours

UNIT 10A.12: Number and algebra 2Elementary set theory9 hours

UNIT 10A.6: Algebra 2Properties of functions; domain and range;mappings; graphs from familiar contextsLinear functionsPlotting straight line equations; parallel andperpendicular lines; points of intersectionSolution of simultaneous linear equations,exact and approximate; regions of linearinequality; interpreting solutions14 hours

UNIT 10A.10: Algebra 3Quadratic functions and their graphsSimple quadratic inequalitiesTangent line to graph of function; maximaand minima; interpretation in physicalsituationSolution of quadratic equations byfactorisation, formula, completing the square14 hours

55%

33 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.1 | Number © Education Institute 2005

GRADE 10A: Number

Ratio and percentage, powers, roots and surds

About this unit This is the only unit exclusively on number for Grade 10 advanced. It builds on the work on number in Grade 9.

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 or extension activities, look at the units for Grade 9 or Grade 11 advanced.

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 use index notation and the laws of indices and write numbers in standard from. They should be able to round numbers to a given number of significant figures. They should be able to solve problems involving fractions, percentages, ratios and proportions and know how to calculate simple interest. They should check their answers and round them to a suitable degree of accuracy.

Expectations By the end of the unit, students will calculate with any real numbers, including powers, roots, and numbers expressed in standard form. They will use proportional reasoning to solve problems involving scale, ratios and percentages, including compound interest.

Students who progress further will be more critically aware of the distinction between approximation and exact value for irrational numbers. They will be fluent in the use of indices. They will understand the construction of the real number system.

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

• graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • irrational, integer, exponent, index, surd, root • decomposition, factor, multiple, divisor, prime number, prime factor • least common multiple (LCM), highest common factor (HCF), lowest

common denominator

UNIT 10A.1 9 hours


Standards for the unit

9 hours SUPPORTING STANDARDS Grade 9 standards

CORE STANDARDS Grade 10A standards

EXTENSION STANDARDS Grade 11A standards

9.2.1 Round numbers and measures to a given number of significant figures.

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

9.2.9 Check answers for accuracy and reasonableness; round answers appropriately. 10A.2.5 Know from definitions that an even number can be written in the

form 2m, where m is an integer, and that an odd number can be written in the form 2n + 1, where n is an integer; understand and use terms such as factor, multiple, divisor, prime, etc.

10A.2.1 Identify the number sets: the set of all real numbers; the set of all integers; + the set of all positive integers {1, 2, 3, 4, …}; – the set of all negative integers; the set of all rational numbers; the set of all non-negative integers, called the set of natural

numbers {0, 1, 2, 3, 4, …}.

10A.2.2 Know when a real number is irrational, i.e. not a member of .

9.2.2 Use index notation and the laws of indices to evaluate expressions with integral powers, including positive and negative powers of 10.

10A.3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key on a calculator.

11A.3.1 Develop further confidence in using the rules for indices.

9.2.4 Use the x2, √x, xy and x1/y keys of a scientific calculator, distinguishing between the root and the decimal approximation.

9.2.5 Find the value of a square root or cube root to a given degree of accuracy using a calculator or spreadsheet.

10A.3.2 Know that a root that is irrational is an example of a surd, as are expressions containing the addition or subtraction of an irrational root; perform exact calculations with surds.

9.2.3 Read and write numbers in the standard form A × 10n, where n is a positive or negative integer and 1 ≤ A < 10; interpret numbers in standard form on a calculator display; use standard form in calculations and to estimate.

10A.3.3 Use standard form in appropriate situations: for exact calculations, to estimate results of calculations and to make comparisons.

9.2.7 Solve problems involving fractions, percentages, ratios and proportions.

10A.3.4 Perform calculations with any real numbers, including mental calculations in appropriate cases.

9.2.8 Calculate simple interest. 10A.3.7 Perform percentage calculations involving taking a percentage of a percentage, inverse percentage, and compound interest.

2 hours

Percentages

2 hours

Compound interest

4 hours

Powers, roots and laws of indices, surds

1 hour

Standard form

10A.5.17 Identify some other common examples of proportional variation.

Unit 10A.1


Activities

Objectives Possible teaching activities Notes School resources

2 hours

Percentages Perform percentage calculations involving taking a percentage of a percentage and inverse percentage.

Review of standard percentage calculations

Class discussion Most students will have a good intuition for the idea of percentage. Ask, for example, what is 10% of a quantity of money or weight, and you will get an immediate and probably correct answer. Pose the following question. • If a shop has an item priced at £100 and there is a subsequent price increase of 10% followed

soon afterwards by a decrease of 10%, is the third price: A. equal to the original, B. greater than the original, or C. smaller than the original?

Expect a response from all students, for example, by taking votes on the three answers. It is likely that some votes will be cast for each option. This motivates careful calculation of the steps involved, and since the numbers are easy they do not distract from the underlying principles. Stress that the key point in the calculations is that the base of the percentage calculation changes from the first step to the second. The guiding principle in all percentage calculations is to focus on the quantity that is the base in the given situation, be it a known quantity at that stage or an unknown.

Revise some groundwork before embarking on more complex percentage problems. For example: • increasing by 7% is equivalent to multiplying by 1.07; • decreasing by 6% is equivalent to multiplying by 0.94.

These ideas are useful for compound interest and appreciation and depreciation calculations.

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

Miscellaneous percentage calculations Revise the various types of question that students should be able to do. These include: • calculation of a percentage increase (given) from a starting value (given); • calculation of percentage increase or decrease from given starting and finishing values; • calculation of percentages equivalent to fractions and vice versa; • two-stage calculations where the base of percentages changes (as in the class discussion).

Stress the importance of well-set-out work to aid both analysis and comprehensibility.

Reverse percentages

Class discussion Consider reverse percentage calculations. Pose a question with a commercial context. • After a decrease of 10%, a price is now £90. What was the original full price?

It is likely that both answers £99 and £100 will get some support. Focus carefully on the problem. Get those who favour £100 to convince those who favour £99. It should be clear to all as a result that a decrease of 10% from £100 produces £90, and an increase of 10% from £90 produces only £99. The only way to settle the issue is to insist that in any change – increase or decrease – the base of percentage calculation is always the original value. Students should realise that this is a reprise of the problem introduced originally.

Unit 10A.1



Exercises Begin with exercises on reverse percentage calculations and then turn to miscellaneous types. Give extra support to less confident students who easily master the reverse percentage process but who lose sight of this when the calculations are mixed.

Compound and simple interest compared

Class discussion At one level, compound interest is just another case of percentage calculation. At another, it is a sophisticated case of the change-of-base problem. Remind students that percentage increase can be regarded as equivalent to a multiplying factor. Highlight the difference between simple interest (already studied) and compound interest by consideration of a simple case of 10% interest compounded annually. The calculations are easy to do but the graphical representation is much more impressive. Use either or both these approaches to make the point clear.

The graph on the left (produced in Autograph) shows the increasing difference between simple and compound interest.

Develop the same graph using spreadsheet software to show the changes dynamically in response to changing the entry corresponding to interest rate.

The spreadsheet compares the effects of simple and compound interest. The interest rate is in cell C1. The other cells are dependent on C1, so, for example, changing it to 20 results in a spectacular change in the comparison. Key definitions for reference:

A2 = 100; A3 = A2 * (1+C$1/100); cells A4 etc. are replicated from A3.

B2 = 100; B3 = B2 + B$2*C$1/100; cells B4 etc. are replicated from B3.

2 hours

Compound interest Perform percentage calculations involving compound interest.

Exercises Set exercises on compound interest and develop students’ ability to apply the idea in different contexts, such as appreciation, depreciation, inflation, price indices. Emphasise that the key technique is to use a multiplier to obtain the new amount after compound interest. If appropriate, develop this into a formula that can be quoted and used in context.

Stress that interest rates can also be negative and that this corresponds to real phenomena such as inflation.

At this level it is not possible to present a technique for accurate determination of (say) a percentage annual interest rate that would result in a specified increase over (say) 5 years, but more able students could calculate this by trial and error on a calculator.

Extensions If time permits, explore the idea of annual percentage rate as a means of comparing competing interest rates on credit cards, for example.

More able students may be interested in the idea of compounding interest at progressively smaller intervals so that

lim 1→∞

⎛ ⎞+⎜ ⎟⎝ ⎠

n

n

xn

emerges. (In this result x represents the fractional interest rate – e.g. 0.05 for 5% per annum – and n the number of times interest is compounded over a year.)



Working with negative indices

Class discussion Ask students to set aside their calculators while they work with index rules.

Begin by reviewing index notation. It has been used in prime factorisations. In that context, it is a shorthand for more cumbersome alternatives, as well as a slick notation for handling the calculation of LCMs and HCFs.

Develop this first into a set of index rules easily justified by experience. Students will probably see these as easy and not particularly valuable. Then pose the issue of meaning for negative indices; by insisting that the rules just established still work, division (where m is less than n) results in a meaning for the negative index m – n.

The same argument works for division of a number by itself, where m = n establishes a meaning for a zero index. Emphasise the meaning of x0. Explain that x0 is undefined when x is zero.

A common misconception of confident students is the erroneous idea that a negative index leads to a negative number. Give them extra help to correct the misunderstanding.

Index rules to be established by experience with positive indices only:

( )

+

−

=

=

=

m n m n

mm n

n

m n mn

x x xx xxx x

Additional rules to establish are:

0 11−

=

=

=mn

nn

mn

x

xx

x x

There is a thorough discussion of this topic at a basic level at descartes.cnice.mecd.es/ ingles/4th_year_secondary_educ_%20A/ Power_%20rational/potencias33.htm.

Exercises The content of exercises should be: • practice of the rules in routine situations; • extension to incorporate zero and negative indices. Calculators should be forbidden.

Comparisons such as 23 and 32 are always instructive and a guard against guesswork.

Working with fractional indices

Class discussion Move on to fractional indices. Use the method of generalisation of existing rules. Almost certainly some students will recognise a power of 1⁄2 as equivalent to a square root if allowed to ponder the problem. This is a difficult idea and it will need time to establish.

Consider explicitly too how to find square and cube roots. In general, for roots that are not obvious, the best advice to give is: • for integers, make a prime factorisation; • for fractions, work with an improper fraction, that is, the ratio of two integers.

It is general practice at this stage to regard fractional powers as single-valued functions where the positive option is preferred when both signs are possible. So 41/2 = +2 and (–27)1/3 = –3, for example.

Students should discover that the processes of taking roots and powers for a fractional index are commutative.

4 hours

Powers, roots and laws of indices, surds Identify the number sets:

the set of all real numbers; the set of all integers; + the set of all positive integers {1, 2, 3, 4, …};

– the set of all negative integers;

the set of all rational numbers;

the set of all non-negative integers.

Know when a real number is irrational, i.e. when it is not a member of .

Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents.

Know that a root that is irrational is an example of a surd, as are expressions containing the addition or subtraction of an irrational root; perform exact calculations with surds.

Perform calculations with any real numbers, including mental calculations in appropriate cases.

Know when an exact solution is appropriate.

Exercises This topic is a difficult one and in many respects counter-intuitive. At this point include: • thorough drill of miscellaneous types; • problems with letters for simplification rather than figures; • problems such as: find x where 9x = 3√3.

Calculators should still be forbidden.

It is essential that students master the concepts and think their way through, since the topic is important as a basis for calculus. They may prefer to use calculators but it is necessary to exclude them in this section of the work.



Using surds

Class discussion The general problem of finding roots has been touched on in the theory of indices and the follow-up exercises. This remaining section looks at the issues that arise in particular cases.

To find a square in a prime factorisation, all the indices double; for perfect squares, a prime factorisation will yield the root immediately – and with corresponding results for cubes and cube roots.

Introduce surds. They are the solution to the problem when there is an odd index, e.g. √(22 × 32 × 5) = 2 × 3 × √5.

Students will no doubt work out √60 = 2 × √3 × √5. This is technically correct. Explain that by convention it is usually written as 2√15.

There is a summary of work at this level on this topic at www.gcseguide.co.uk/surds.htm.

Some work on surds also occurs in Unit 10A.3, Algebra 1.

Exercises Give students practice before moving on. The practice should include: • finding exact roots by using surds; • simplification with surds (which brings out the obvious but subtle point that (√2)2 = 2, and other

such results).

Continue to ban calculators. Less confident students will not appreciate the value of an answer such as 2√15 in preference to a decimal approximation.

Introducing irrationals

Class discussion Up to now, students will have used a calculator to evaluate √60, and may have little feel for the difference between an approximation (to a number of decimal places) and an exact value. As a rough guide, explain that a surd answer is always correct but if an answer in a practical application is required a further decimal approximation should be given. Explain also that test questions occasionally deviate from this rule.

The main point to reach is the concept of irrational number. Judgement is needed on how far to go at this stage. Inform students at a minimum that roots of primes are termed irrational and their principal distinguishing feature is that they have infinite non-repeating decimal expansions.

This is not a new idea since it has been met when dealing with π. Once you have made that point, students will easily see that 5√2 is also irrational (since multiplying a non-repeating decimal by an integer will not convert it into a repeating one).

Students will then be able to begin to distinguish from .

Exercises Give students some practice to establish the distinction between the different number sets.

A Venn diagram will help to bring out the way the real number system fits together.

Extension

As an extension for the most able students, extend work to include the proof of the irrationality of √2 and the rationality of finite and repeating decimals, and to conclude that irrationals certainly exist and cannot have repeating decimal expansions. For further scope of irrational numbers, see mathworld.wolfram.com/IrrationalNumber.html.



Standard form for small numbers

Class discussion Standard form has already been met in Grade 9 for numbers greater than 10 (so that the index in standard form is positive).

Extend this to numbers that are less than 1 (and trivially those between 1 and 10); this should now be quite simple. Take the opportunity to reinforce the idea of a zero index.

Students should use calculators during the discussion and in exercises.

Check that students can use the exponent key of a calculator to match standard form notation to a calculator’s display.

1 hour

Standard form Use standard form in appropriate situations: for exact calculations, to estimate results of calculations and to make comparisons.

Work to expected degrees of accuracy.

Understand exponents and nth roots; use the xy key on a calculator.

Exercises Include the following in exercises: • turning ordinary (large and small) numbers into standard form; • putting numbers into a calculator using standard form notation; • interpreting calculator answers in standard form (e.g. turning notation such as 2.62E15 into

2.62 × 1015); • finding squares and roots (using the xy key); • the four arithmetic operations on numbers in standard form (especially in context); • appropriate accuracy for calculated answers.


Assessment

Examples of assessment tasks and questions Notes School resources

Without using a calculator, evaluate (53)4 ÷ 510.

Use a calculator to evaluate 79.

Simplify 81/3 × 2–1.

Calculate (√2 – 1)(√3 – √2).

To four significant figures, the speed of light is 299 800 000 metres per second. Write this in standard form.

The Earth is approximately a sphere of radius 6378 kilometres. Without using a calculator estimate the circumference at the equator.

The mass of the Earth is 5.98 × 1024 kg. A typical man has a mass of about 70 kg . Approximately how many men would have a total mass equal to that of the Earth?

Light travels at about 300 000 kilometres per second. Use standard form to find the distance away from the Earth of a light-emitting body whose light signal is received at Earth one year after it is emitted.

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 to incorporate in the activities.

The Earth completes its orbit around the Sun in 365 days. The Earth is 148.8 million kilometres from the Sun. Assume that the Earth’s orbit is circular and that it travels around the Sun with constant speed. Calculate the Earth’s speed in kilometres per hour.

After Haya’s salary is increased by 15% and Abdullah’s salary is decreased by 27%, Haya and Abdullah both end up with an annual salary of QR 72 000. What were their original salaries? What percentage of Abdullah’s original salary was Haya’s original salary?

The diagram shows water flowing through some pipes. The water starts at A. At each junction the percentage of the inflowing water flowing out through the pipes is indicated. What percentage of the original water flows out at B? What percentage flows out at C?

Owing to inflation, the price of a television in a store is increased by 15%. In the sales at the end of the year, the price is then reduced by 15%. Does the television revert to its original pre-inflation price? Or is it more, or less? Explain your reasoning.

QR 100 000 has to be invested in deposit accounts. There is a choice of two accounts. One account pays an annual interest of 4.6%. The other account pays interest of 1.5% three times per year. What is the AER of the second account? Which is the better account to invest in and how much more interest will there be after one year in this account than in the other account?

Unit 10A.1

41 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.2 | Geometry 1 © Education Institute 2005

GRADE 10A: Geometry 1

Angle properties, congruent triangles and proof

About this unit This is the first of two units on geometry (of the suite of four on geometry and measures) for Grade 10 advanced. It builds on the work on geometry in Grade 9. Its main purpose is to give students a clear idea of what is involved in mathematical proof and so to lay the foundations for further work.





Previous learning To meet the expectations of this unit, students should already be able to deduce properties in a given plane figure, using their knowledge of angles and properties of shapes. They should know the conditions of congruence and similarity and be able to determine whether two triangles are congruent or similar. They should be able to develop a simple proof.

Expectations By the end of the unit, students will use their knowledge of geometry to solve practical and theoretical problems relating to shape and space. They will break down complex problems into smaller tasks, and develop and explain short chains of reasoning. They will generate simple mathematical proofs, and identify exceptional cases. They will calculate the interior and exterior angles of regular polygons. They will understand and use congruence and similarity to solve problems and to prove results.

Students who progress further will be able to apply established results to multi-step proofs. They will identify similar triangles and their corresponding angles and sides.

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

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

• computers with Internet access and dynamic geometry software for students

• sharp pencil, protractor and ruler for each student

Key vocabulary and technical terms Students should understand, use and spell correctly: • triangle, corresponding, alternate, proof, ratio, perpendicular, bisect,

intersection, isosceles, rhombus, hypotenuse, right angle • similar, congruent

UNIT 10A.2 10 hours






9.1.2 Solve more complex problems by breaking them down into smaller tasks.

10A.1.4 Break down complex problems into smaller tasks. 11A.1.4 Break down complex problems into smaller tasks.

10A.1.6 Develop short chains of logical reasoning, using correct mathematical notation and terms.

11A.1.6 Develop chains of logical reasoning, using correct mathematical notation and terms.

9.1.4 Explain and justify the steps taken to solve a problem or arrive at a conclusion, orally and in writing.

10A.1.7 Explain their reasoning, both orally and in writing. 11A.1.7 Explain their reasoning, both orally and in writing.

9.1.6 Develop a simple proof. 10A.1.8 Generate simple mathematical proofs, and identify exceptional cases. 11A.1.8 Generate mathematical proofs, and identify exceptional cases.

9.1.7 Generalise where possible and identify exceptional cases.

10A.1.9 Generalise whenever possible. 11A.1.9 Generalise whenever possible.

9.5.1 Use knowledge of angles and properties of 2-D shapes to conjecture or deduce properties in a given plane figure.

10A.6.1 Use knowledge of angles at a point, angles on a straight line, and alternate and corresponding angles between parallel lines and a transversal line to present formal arguments to establish the congruency of two triangles.

9.5.4 Identify congruent triangles and their corresponding angles and sides; know the conditions of congruence and determine whether two triangles are congruent.

10A.6.2 Establish the congruency of two triangles to generate further knowledge and theorems about triangles, including proving that the base angles of an isosceles triangle are equal and that the line joining the mid-points of two sides of a triangle is parallel to the remaining side.

9.5.5 Use the properties of congruence or similarity of triangles to solve problems, e.g. find unknown sides or angles of similar or congruent triangles.

10A.6.3 Understand similarity of two triangles and other rectilinear shapes, knowing that similarity preserves shape and angles, but not size; make inferences about the lengths of sides and about the areas of similar figures; prove that if two triangles are similar, then the ratio of the areas of the two triangles is the square of the ratio of any pair of corresponding sides of the two triangles chosen in the same order; in three dimensions, calculate the ratio of the volume of a scale model to the volume of the actual object.

2 hours

Angle properties

6 hours

Congruent triangles and their uses

2 hours

Similar shapes

10A.6.4 Calculate the interior and exterior angles of regular polygons; name polygons with up to ten sides.

Unit 10A.2


Activities


Ground rules for geometry

Class discussion Students have constructed parallels using a set square in earlier grades. Remind them how to do it. They have also met the result that vertically opposite angles are equal.

These two pieces of knowledge combined build up the angle properties of parallels and a transversal. Formalise this so that the terminology is made clear.


Make sure that students understand and can use the terms: • corresponding angles; • alternate angles.

Revise also: • supplementary angles; • complementary angles.

Establish and use the convention of naming angles with small letters and points with capital letters.

Students should also be able to use the conventional method of specifying an angle with three letters for points (so that ∠RQS is a2, for example). Point out that sometimes identical small letters with distinguishing suffices are used to denote angles that are equal.

These properties are reviewed in Keymath (www.keymath.com/DG/dynamic/ parallel_lines.html)

2 hours

Angle properties Break down complex problems into smaller tasks.

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

Explain their reasoning, both orally and in writing.

Generate simple mathematical proofs, and identify exceptional cases.

Generalise whenever possible.

Calculate the interior and exterior angles of regular polygons; name polygons with up to ten sides.

Develop a set of abbreviations to use in arguments, such as: • the angle sum of a triangle is 180° (∠ sum of ); • vertically opposite angles (vert. opp. ∠s); • angles at a point add to 360° (∠s at point); • angles on a straight line add to 180° (∠s on straight line).

Practise using the abbreviations. Derive the results about interior and exterior angles in a polygon.

Extension For more able students, refer to Euclid’s axioms. There is a discussion of these axioms at Mathworld (mathworld.wolfram.com/ EuclidsPostulates.html).

In the diagram (of a pentagon) starting at A, the external angles are marked a1, b1, c1, d1, e1. Small parallels to the side EA have been drawn from B, C, D, so that all the angles marked with a are equal as corresponding angles. Similarly, small parallels to AB have been drawn from C, D and E, so that all the angles marked b are equal.

If this process is continued, the result at one vertex is a full 360° made up of a, b, c, d and e. In consequence the external angles of a pentagon make up 360°.

Students will readily see – if necessary by drawing, say, a hexagon – that the argument generalises to a polygon of any size.

Show by using a diagram that the argument still holds if one of the angles is reflex.

Unit 10A.2



Demonstrate this result informally by marking the figure on the floor big enough for a student to walk along the edges. In the process of doing this the student will have turned round exactly once. This is a good informal approach but should be accompanied by the formal approach to demonstrate the use of logical argument.

Ask students to prove for themselves the corresponding result for the sum of the internal angles.

Another way of looking at this result can be found at www.ies.co.jp/math/java/ samples/gaikaku.html.

Exercises Give students some problems or puzzles based on the angle properties just revised or established. Encourage the use of the proposed abbreviations to justify the arguments.

Students will master these ideas easily, but less confident students may have difficulty with the justification of steps in an argument. Give them extra support to develop their confidence.

With students, prove the exterior angle property of a triangle.

Examples • Find the exterior angle of a regular

polygon with ten sides. • List the possible numbers of sides that

a regular polygon can have if its external angle is an integral number of degrees.

• Find all the unmarked angles in the figure below.

6 hours

Congruent triangles and their uses Use knowledge of angles at a point, angles on a straight line, and alternate and corresponding angles between parallel lines and a transversal line to present formal arguments to establish the congruency of two triangles.

[continued]

Congruent triangles and the properties of an isosceles triangle

Class discussion Students should be familiar with the four tests for congruence: • three equal sides (SSS); • two corresponding sides and the included angle (SAS); • two angles and the corresponding side (AAS); • right angle, hypotenuse and side (RHS).

Rehearse these as a basis for what is to follow. They derive in practical terms from the minimal sets of information required to construct a unique triangle. Draw attention to the connection of these constructions to the four tests.

Just as with the properties of parallels and a transversal, these results are axiomatic.

For revision of congruence and similarity at this level see GCSE Bitesize Revision (www.bbc.co.uk/schools/gcsebitesize/ maths/shapeh).

Other geometry materials related to this subject matter are at MathsNet (www.mathsnet.net/geometry/index.html). Among the other things on offer is work on Euclid’s elements and constructions.



[continued]

Establish the congruency of two triangles to generate further knowledge and theorems about triangles, including proving that the base angles of an isosceles triangle are equal and that the line joining the mid-points of two sides of a triangle is parallel to the remaining side.

The two results specified in the standards are examples of the easy and hard types of application. In each case, the result must stem from a clear statement of the starting point – what is ‘given’. Translate that starting point into a diagram labelled sufficiently clearly and comprehensively to make the argument easy to follow.

Given: a triangle ABC in which AB = AC.

To prove: ∠ABC = ∠ACB.

Proof: Construct the line that bisects ∠BAC (so p1 = p2). In ABD and ACD we have: AB = AC (given) AD is common p1 = p2

So the triangles are congruent by SAS. It follows that corresponding angles and sides are equal; in particular the angles required to be proven are equal.

Corollaries: • BD = DC (so D is the mid-point) • The two equal angles at D are right angles.

Extension for more able students This result can also be proved by showing that ABC and ACB (note the order of the letters) are congruent. This is more subtle.

Students will be familiar with the elementary properties of rectangles, squares and other quadrilaterals. Review the definitions. Show how these familiar properties can all be established by congruent triangle arguments.

To prove that the line joining mid-points Q and T is parallel to the base, proceed as follows.

Draw parallels from Q and T until they meet at U, not necessarily on RS.

Now prove that PQT and UTQ are congruent (AAS), as are RQU and TUQ (SAS), and SUT and QTU (SAS).

Then the three angles at U making up ∠RUS are one from each vertex of PQT and so form a straight line. Hence U is identical to U′ and lies on RS. QT is parallel to both RU and US by equal corresponding angles.

Use a dynamic geometry system such as Geometer’s Sketchpad or Cabri to explore this theorem. Students may not accept at first that U and U′ are to be considered as separate points.

The application measures the marked angles made by the side with the base and with the transversal (which connects the sides’ mid-points).

Drag the top point to see that the angles stay equal as they vary. Equal corresponding angles imply that the two lines are parallel.

Extension Set more able students similar challenges, such as: • if QT is parallel to the base, and Q is a mid-point, then so is T; • if Q, T and U are mid-points, then QTU is similar to SRP.



Exercises Avoid problems in which angles have to be calculated; concentrate on proofs. Use problems that involve tracking equal angles round a diagram, using known properties. Stress and insist on the routine: • set out what is given; • draw a good diagram, properly lettered; • use small suffixed copies of the same letters to mark equal angles as the argument develops; • at each step give a reason; • the existence of an abbreviation for reference is a good indication that the reason is an acceptable

one to give.

Base the exercises on elementary properties of quadrilaterals. These have plenty of scope for simple one- or two-stage arguments and will give students confidence that they can succeed.

Encourage students in exercises such as these to discover and list properties of the different quadrilaterals, and to see how later results depend on those already established.

Ask for a minimal definition of any suggested quadrilateral. A square, for example, is: • a quadrilateral with four sides equal and one right angle; • a quadrilateral with two adjacent sides equal and three right angles; • a quadrilateral with two equal diagonals which intersect at right angles.

Examples • In a parallelogram ABCD, prove that triangles ABD and CDB are congruent. Use that result to write

down as many properties of a parallelogram as you can. • Prove that the diagonals of a parallelogram intersect at their mid-points. (Hint: draw the figure;

which triangles are congruent?) You can use properties established in the last question. • If a triangle ABC has a mid-point X for the side BC, and a line is drawn parallel to BA through X,

show that the line intersects AC at its mid-point.

Allow good time for these exercises. They are the foundation for future studies, not just in geometry.



2 hours

Similar shapes Understand similarity of two triangles and other rectilinear shapes, knowing that similarity preserves shape and angles, but not size; make inferences about the lengths of sides and about the areas of similar figures; prove that if two triangles are similar, then the ratio of the areas of the two triangles is the square of the ratio of any pair of corresponding sides of the two triangles chosen in the same order; in three dimensions, calculate the ratio of the volume of a scale model to the volume of the actual object.

Areas and volumes of similar shapes

Class discussion Students will be familiar with ideas of similarity. They may mix the vocabulary of congruence and similarity even if the ideas are quite distinct. Work steadily towards correct and precise use of language.

Once students have mastered the idea of similarity, they can appreciate an application in biology. Modelling animals for this purpose as similar in shape, comparing a dormouse with an elephant shows that their volumes (related to their consumption of energy and production of heat) are scaled up much more than their surface areas (related to their heat-loss factors). Hence a large animal has greater chance of survival in cold conditions.

The new idea here is one that is counter-intuitive. Students may forget it when they meet it out of context. To combat this, constantly go back to the simple example of an arrangement of squares. If one square has an area of one unit, four congruent non-overlapping squares arranged in a 2 by 2 square have an area of four square units, nine squares arranged in a 3 by 3 square have an area of nine square units, and so on.

Circle formulae show how the same results apply to a figure that is not a square or rectangle. If the radius of a circle is doubled, its circumference is also multiplied by 2, but its area by 4, and so on.

Demonstrate the result for volume in the same way by using cubes. If students are familiar with the results for areas and volumes of spheres, use those as examples too.

The results in this section can be proved for rectangles and for triangles. An indication of how they can be proved for more complex figures by breaking them down into smaller parts can be discussed. The circle shows that the argument is still more general, but the general principle cannot be established at this level without recourse to limits.

There is a wider context for this topic set at the website Maths GCSE Guide (www.gcseguide.co.uk/area_and_volume.htm).

Exercises Split this into two sessions: • one on areas; • one on volumes.

Advise students to write down the scale factors that they use so that they can focus on whether they relate to linear measurements, areas or volumes.

Bring in also the question of unit conversions – for example, how many cubic millimetres there are in a cubic centimetre. This is a frequent source of error yet the methods of this section show how to visualise the problem.

The exercises can be used as an opportunity to revise standard form.


Assessment


Prove that each of the angles in an equilateral triangle is 60°.

A goldsmith has a block of gold in the shape of a cube. He wants to make another gold cube that has exactly twice the volume of the first cube. What scale factor must he use?

Two similar-shaped gas-filled balloons are made of a special material. The area of material used in one balloon is 100 cm2. The material for the other balloon has an area of 225 cm2. Calculate the ratio of the volume of the larger balloon to the volume of the smaller balloon. Give this ratio in its simplest form.


The diagram shows two triangles ADE and BCE. Side AD is parallel to side BC. Explain why the two triangles are similar to each other.

Calculate the missing lengths for triangle BCE.

Calculate the length of CD in the diagram on the right.

The interior angle of a polygon is 160°. How many sides does the polygon have?

A scale model of a dhow has a volume of 300 cm3. The length of the actual dhow is 100 times longer than the length of the model. What is the volume of the dhow? Give the answer in appropriate units.

A model of a mosque is built to a scale of 1 : 200.

a. The mosque is 50 metres high. Find the height of the model in centimetres.

b. The angle of elevation of the top of the mosque from the main entrance is 17.3°. What is the corresponding angle for the model?

c. The model has a roof area of 2.5 square metres. Calculate the roof area of the mosque.

Unit 10A.2

49 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.3 | Algebra 1 © Education Institute 2005

GRADE 10A: Algebra 1

Equations, identities and formulae

About this unit This is the first of three units on algebra for Grade 10 advanced. It builds on the work on algebra in Grade 9. The focus of this unit is algebraic manipulation.





Previous learning To meet the expectations of this unit, students should already be able to add, subtract, multiply and divide algebraic fractions, and write and solve linear equations. They should be able to expand the product of two simple linear expressions and factorise the corresponding product by removing common factors. They should be able to change the subject of a simple formula.

Expectations By the end of the unit, students will be aware of the role of symbols in algebra. They will generate and manipulate algebraic expressions, including algebraic fractions, equations and formulae. They will multiply any two polynomial expressions and factorise quadratic expressions. They will use algebraic methods to solve linear equations.

Students who progress further will recognise the role of a surd in irrational roots of integers. They will have no difficulty using the difference of squares to rationalise a denominator that is linear in a surd. They will solve algebraic equations with confidence. They will use factors with confidence in all contexts where they are appropriate for simplification or solution.

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

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

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

Key vocabulary and technical terms Students should understand, use and spell correctly: • words related to surds: root, surd, rational, irrational • words related to algebraic manipulation: rationalise, rationalisation,

simplify, simplification • words related to algebraic reasoning: proof, solve

UNIT 10A.3 14 hours






10A.2.2 Know when a real number is irrational, i.e. when it is not a member of .

11A.2.1 Make appropriate use of knowledge of number sets.

9.3.4 Add, subtract, multiply and divide algebraic fractions.

10A.3.5 Add, subtract, multiply and divide any two fractions and understand how to use a unit fraction as a multiplicative inverse.

10A.4.1 Solve any linear equation with one unknown. 9.3.6 Write and solve linear equations, including simple cases of fractional linear equations, and apply these skills to solving problems; verify the solution.

10A.4.7 Develop confidence and accuracy in working with symbols, understanding that the transformation of all such algebraic objects generalises the well-defined rules of arithmetic. Recognise that letters are used to represent: • the solution set of initially unknown numbers in

equations; • defined variables in formulae; • generalised independent numbers in identities; • new equations, expressions or functions in terms of

known, or given, expressions or functions.

9.3.1 Expand expressions of the form a(x ± b), (x + a)2, (x – a)2, (x + a)(x – a), (x ± a)(x ± b), where a and b are positive or negative integers.

10A.4.8 Use brackets and correct order of precedence of operations when performing numerical or algebraic calculations.

10A.4.9 Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions; multiply any two polynomials, collecting up and simplifying similar terms.

9.3.3 Factorise algebraic expressions: • by removing common factors from expressions

such as ax ± ay and ax + bx + ay + by; • by recognising the factors of expressions such

as a2x2 − b2y2 and x2 ± 2ax + a2; where a and b are positive or negative integers.

10A.4.10 Factorise quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions.

10A.4.12 Simplify numeric and algebraic fractions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

9.3.5 Change the subject of a simple formula. 10A.4.13 Generate formulae from a physical context, and rearrange formulae connecting two or more variables.

2 hours Algebra with surds

2 hours Manipulation of identities

2 hours Factorising trinomials

2 hours Special cases: squares of sums and differences of squares

2 hours Long multiplication and more complicated simplification

2 hours Simplifying algebraic fractions 2 hours

Equations and formulae

10A.4.14 Substitute an expression for a given variable into a different formula containing this variable.

Unit 10A.3


Activities


Working with surds

Class discussion The practical purpose of surds has been covered in unit 10A.1. This unit focuses on manipulating surds in context.

Simplification of expressions involving surds requires two skills: • the ability to move from an expression like √12 to the equivalent 2√3; • the realisation that expressions like 5√3 – 2√3 simplify just as 5x – 2x does.

Work several examples in discussion with the class to show how these principles can be used.

Explain to students that, at the end of their course of study, they will be able to answer questions such as: • How can you know that π is irrational? • How can you know that the expansion of π does not recur?

Examples • Express √216 as simply as possible, using a

surd if required. • Simplify 2√2 + 5√8.

• Simplify 3 7 22

− .

• Rationalise the denominator in 32 7

.

A summary of work at this level on this topic is on the Maths GCSE Guide website (www.gcseguide.co.uk/surds.htm).


Exercises Give students practice in the use of surds. Include: • expressing roots of integers in terms of the smallest possible surds, to consolidate the work

of unit 10A.1; • simplifying expressions which involve surds; • arithmetic operations which include the appearance of surds as factors in the denominators. For these exercises, ban calculator use since exact answers will be required. Conclude by checking with a worked example that the case of denominators like a + b√n has been fully mastered.

Extension for more able students An example of a harder case is:

3 2 23 5 2

3 2 2 3 5 23 5 2 3 5 229 21 2

41

+−

+ += ×− +

+=−

2 hours

Algebra with surds Know when a real number is irrational, i.e. when it is not a member of .

Rationalise a denominator of a fraction when the denominator contains simple combinations of surds.

Problem solving Give students a problem to solve such as: • Find the polynomial p(x) with integer coefficients such that one solution of the equation

p(x) = 0 is 1 + √2 + √3. The solution can be found by forming the equation x – 1 = √2 + √3 and squaring both sides.

Unit 10A.3



Working with identities

Class discussion Return to basic algebraic operations. Introduce identities that incorporate expansions of double brackets. The example given (and others of the type) leads to problems associated with priority of operations and signs.

Problem cases to cover include: • (3x)2 = 9x2; • the difference between (–x)2 and –x2; • obtaining the correct signs in an expression such as –5(3t – 4u)(7t + 6u).

Explore the difference between identities and equations – almost certainly implicitly understood by students but never formally considered. Encourage students to work down the page rather than across – as in the examples in the notes for this unit. This helps students to appreciate that: • an equation has an equals sign = in the middle and a logical connector such as ⇒ at the

beginning of each line; • an identity has a single equals sign at the beginning of each line. Ask questions such as: • Is (x + 4)2 = x(x + 12) – 4(x – 4) an equation or an identity? Explain your reasoning.

Presentation for identities:

2 2

2 2

2(3 4)(5 7) 5 (6 1)2(15 1 28) 30 530 2 56 30 53 56

s s s ss s s s

s s s ss

+ − − −= − − − += − − − += −

Presentation for equations:

2 2

2 2

2(3 4)(5 7) 5 (6 1)2(15 1 28) 30 5

30 2 56 30 53 56

563

s s s ss s s ss s s s

s

s

+ − = −⇒ − − = −⇒ − − = −⇒ =

⇒ =

2 hours

Manipulation of identities Develop confidence and accuracy in working with symbols, understanding that the transformation of all such algebraic objects generalises the well-defined rules of arithmetic. Recognise that letters are used to represent: • the solution set of initially

unknown numbers in equations;

• defined variables in formulae;

• generalised independent numbers in identities;

• new equations, expressions or functions in terms of known, or given, expressions or functions.

Use brackets and correct order of precedence of operations when performing numerical or algebraic calculations.

Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions; multiply any two polynomials, collecting up and simplifying similar terms.

Class exercises Give students: • examples requiring the simple expansion of brackets; • more complicated expressions.

Include also cases of identities to be established where both the left-hand side and right-hand side have to be simplified. These pave the way for trigonometric identities of the same kind that come later.

Impress on students who take readily to the ideas that lining up equals signs in the middle of the page or column improves the presentation of the work.

For example: • Simplify (2x + 3)(5x – 3) – (4x + 3)(5x – 3). • Prove that

(x + 2)2 – (x – 2)2 = 2[(x + 1)2 – (x – 1)2].

Here is a model solution for the second example.

2 2

2 2

2 2

2 2

2 2

2 2

LHS ( 2) ( 2)4 4 ( 4 4)4 4 4 4

8RHS 2[( 1) ( 1) ]

2[ 2 1 ( 2 1)]2[ 2 1 2 1]2 48

LHS RHS, as required

x xx x x xx x x x

xx x

x x x xx x x x

xx

= + − −= + + − − += + + − + −== + − −= + + − − += + + − + −= ×=

⇒ =

In the above example, some students may spot the possibility of using the difference of squares.



2 hours

Factorising trinomials Factorise quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions.

Finding factors for trinomials

Class discussion Present this as a reversing of a previous problem. Students have practised the expansion of products of brackets such as (x + 2)(x + 3).

Give them an expression such as x2 – 3x – 4. Explain that this is the answer when two brackets were multiplied out. This is the key to finding the factorisation, since it depends on a sophisticated form of guesswork rather than a constructive approach.

Consider the simplest cases first where only one letter is present and the coefficient of the square is 1. Use simple cases to show how the integer coefficients in the factors must arise and how the signs in the problem posed can be used to determine them.

Students will soon discover a ‘rule’ such as: in factorising x2 + bx + c look for two numbers which multiply to give c and which add to give b. This is an approach to note but not to approve, because it does not easily generalise to the case ax2 + bx + c.

Include in the simple cases examples such as 3 + 2x – x2.

These simple cases can be illustrated geometrically so that an abstract process can still be given a concrete justification and representation.

Move on to consider cases where the coefficient of the square term is not 1. There are many approaches to this. The best is to work systematically in stages: • discover and separate any common factors; • look for the elements which are certain in the answer; • systematically list the possibilities for the uncertain elements; • eliminate any of the possibilities which would reintroduce common factors; • find the solution by trial and error from the options which remain.

This approach is illustrated here.

2

2

12 10 12

2(6 5 6)

1 6 possible (1)6 1 NO: 6 would be a common factor2(6 )( )2 3 NO: 2 would be a common factor3 2 NO: 3 would be a common factor

1 6 NO: 2 would be a common factor6 1 NO: 3 wouOR 2(3 )(2 )2 33 2

x x

x x

x x

x x

− −

= − −

=

ld be a common factorpossible (2)NO: 3 would be a common factor

2(3 2)(2 3)x x= + −

There are only two possible combinations of integers that need be considered once common factors have been ruled out. Trial and error shows which of these it has to be.

An interactive web page on this technique is on the website New Signpost Mathematics (www.media.pearson.com.au/schools/cw/ au_sch_mcseveny_nsm9_5153_1/dnds/ 11_tri1.html).



Exercises Give students practice in factorising quadratic expressions, working from easier to harder examples.

Make sure that students check their solutions, as plausible but wrong answers can easily be obtained. Make checking part of the process of solution, as with simultaneous equations.

2 hours

Special cases: squares of sums and differences of squares Factorise quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions.

Squares of sums and differences

Class discussion Students should have no difficulty in expanding (a + b)2 and (a – b)2. Compare and contrast the two answers. Emphasise that each answer is more than just a2 + b2. Use a diagram to illustrate.

Consider the corresponding question of (x + y)(x – y).

More able students will remember this if they have seen it before since the simple answer is intriguing; less confident students will certainly need to work it out, and may need support in handling signs. Again a diagram will help.

Use the applet Proof without words – Completing the square (illuminations.nctm.org/tools/index.aspx).

Also try the three applets Geometric algebra 2D – Problems 1, 2 and 3 (www.fi.uu.nl/wisweb/welcome_en.html).

Use this kind of diagram to illustrate simple algebraic identities, especially in the early stages of the work.

Exercises Use straightforward applications of the types just considered. Aim to consider the skills of expanding squares of binomial sums and differences and recognising the patterns as an aid to factorisation.

Add variety by working in applications in arithmetic.

Examples

Evaluate these without a calculator. • 552 – 452 • 0.7632 – 0.2372



2 hours

Long multiplication and more complicated simplification Multiply combinations of monomial, binomial and trinomial expressions; multiply any two polynomials, collecting up and simplifying similar terms.

Algebraic long multiplication

Class discussion Consider ‘long multiplication’ of algebraic expressions.

Approach this as repeated application of the distributive law. Present this fully first, to show how it will produce an accurate answer, possibly with several lines of working.

As a bridge to the formal ‘long multiplication’ method, use a grid.

Students will have experienced various methods of keeping tally of progress, such as lightly crossing out terms as they are used in simplification. Show how this is still a valid method of ensuring accuracy.

Once these ideas are established, introduce the alternative ‘long multiplication’ approach. This method is worth seeing not just because it is useful but because it can later be invoked to demonstrate the formation of Pascal’s triangle (in Grade 11).

The ‘long multiplication’ approach has unlimited scope for further development, such as with three or more factors (and in demonstrating later how the binomial theorem works).

Examples This method works in simple cases.

2

2 2

3 2 2

3 2

(2 3)( 5 7)2 ( 5 7) 3( 5 7)2 10 14 3 15 212 13 29 21

x x xx x x x xx x x x xx x x

− − += − + − − += − + − + −= − + −

As a bridge to the method below, use a grid.

x2 –5x +7

2x 2x3 –10x2 +14x

–3 –3x2 +15x –21

2x3 -- 13x2 + 29x – 21

The method below is more generally useful. It gives a better insight into the structure of the answer and how it arises.

x2 – 5x + 72x – 3

2x3 – 10x2 + 14x– 3x2 + 15x – 21

2x3 – 13x2 + 29x – 21

Exercises Get students to practise the technique.

Include interesting examples, such as (1 + x + x2 )(1 – x) = 1 – x3.

Extend to examples which use these techniques but which also: • bring in surds; • require the establishment of identities; • work the solution of equations.

Encourage more able students to master and remember the identities: • a3 + b3 = (a + b)(a2 – ab + b2) • a3 – b3 = (a – b)(a2 + ab + b2)



2 hours

Simplifying algebraic fractions Simplify numeric and algebraic fractions by cancelling common factors.

Add, subtract, multiply and divide any two fractions and understand how to use a unit fraction as a multiplicative inverse.

Cancelling in algebra

Class discussion

The key aim of this section is to transfer skills from arithmetic to algebra. Students should do this readily in the case of simple fractions, of which they already have considerable experience. They will not so easily see that cancellation skills applied to numbers and then to simple multiples of letters can be applied also to longer algebraic expressions (i.e. brackets).

Common errors and misunderstandings to look out for include attempts to cancel expressions

such as 2a ba b++

. Deal with these misunderstandings by returning to the justification for the

simplest forms of cancelling in cases like 6 3 2 38 4 2 4

×= =×

.

Stress that underlying any form of cancelling is a prior exercise in factorising. Use interesting

cases like 3 1

1tt

−−

or 2 1

1tt

−+

to justify the effort.

Exercises There is plenty of material in textbooks to support practice. It is essential to do enough but not too much.

Limit the practice at this stage to the simplification of fractions, leaving multiplication and division for the next step.

More on cancelling

Class discussion Move on to the next stage of the work: extension to the multiplication and division of fractions, involving factorisation into brackets and powers of brackets as well as into powers of letters. There is nothing new here beyond the elaboration of the same techniques. The importance of factorisation is the key – emphasise it again and again.

A point which students will probably have met but not necessarily fully appreciated in the preceding work is that, in a case like the example on the right, factors such as (2 – x) and (x – 2) are essentially equivalent for the purposes of cancelling.

For x ≠ 2: 2

2

2 82

2( 4)2

2( 2)( 2)2

2( 2)( 2)( 2)

2( 2)

xx

xx

x xx

x xx

x

−−

−=−− +=

−− +=− −

= − +

Class exercises Again, the selection available in texts is extensive. Provide enough to give confidence but not to the extent of boredom.



Substitution

Class discussion and exercises The work so far in this unit has emphasised explicitly the equivalence of letters and brackets in the algebraic structure of expressions. 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.

Take the opportunity to revise the solution of linear equations with one unknown.

2 hours

Rearranging formulae: Class 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 can now be made 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 which 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) ( 1); find .( 1)

(1 )

1 1

x x y y xx x y x xy y

x xy yx y y

y yxy y

= − ≠= − ⇒ = −

⇒ − = −⇒ − = −

−⇒ = =

− −

Examples

Find R in terms of R1 and R2 when

1 2

1 1 1R R R

= + .

Class exercise and discussion

Routine practice of this type of question should follow. The time spent 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 that 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.

The three different edges of a solid cuboid have lengths x, 2x and y, as shown. All the lengths are measured in centimetres. The total surface area of the cuboid is 800 cm2. Find a formula for y in terms of x. What is the total length of all the edges of the cuboid? Give the answer in terms of x.


Assessment


Is (x – a)(x2 + ax + a2) = x3 – a3 an equation or an identity? Discuss what happens to this mathematical statement when a is replaced throughout by –a.

Without using a calculator, find the exact value of:

a. 7.922 – 2.082 b. (√5 + √3)(√5 – √3)

Rationalise the expression 2 .1 3+

Simplify the expressions:

a. (2x – 3)(x2 + x – 10) b. 3 2 2 3

2 2a b a b

a b− .

Melons cost QR 1.5 each and apples cost QR 3.75 per kilogram. A woman buys apples and melons at the supermarket. Set up a formula to describe the total cost of her purchase. Investigate how many melons and how many kilograms of apples she could buy for QR 30.


The volume of a solid cylinder of length h and radius r is V. Find a formula for the curved surface area, A, of the cylinder in terms of r and h. Use this formula to find a formula expressing V in terms of A and r.

Make x the subject of the formula a b c

x b−= .

MEI

Solve:

a. 5(3 7) 12 4x x− = + b. 5 608 p

= c. 300 45w

=+

MEI

a. Factorise and simplify the expression (x – y)2 – y2.

b. By putting x = 1⁄2 into the expression in (a), calculate the exact value of (991⁄2)2.

MEI

Unit 10A.3

59 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.4 | Statistics 1 © Education Institute 2005

GRADE 10A: Statistics 1

More advanced diagrams and ICT

About this unit This is the first of two units on statistics for Grade 10 advanced. It builds on the work on data handling in Grade 9.





Previous learning To meet the expectations of this unit, students should already be able to identify questions and answer them by collecting, organising, representing, analysing and interpreting data. They should be able to construct frequency diagrams, choosing equal class intervals. They should be able to estimate and calculate statistics for small sets of discrete or continuous data, including grouped data.

Expectations By the end of the unit, students will distinguish between qualitative or categorical data and quantitative data, and between discrete and continuous data. They will be able to locate sources of bias. They will plan questionnaires and surveys to collect meaningful primary data from representative samples. They will group data and plot histograms and other frequency and relative frequency distributions. They will draw stem-and-leaf diagrams and box-and-whisker plots. They will pose and solve problems in a range of contexts, breaking down complex problems into smaller tasks, and synthesising and interpreting information in various forms.

Students who progress further will use a calculator with statistical functions, and ICT packages, to aid analysis of large data sets and to present statistical tables and graphs.



• spreadsheet software such as Microsoft Excel • computers with Internet access, spreadsheet and graph plotting software

for students • graphics calculators and scientific calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • data, qualitative, categorical, quantitative, discrete, continuous, primary,

secondary • range, percentile, interquartile range, semi-interquartile range, mode,

modal class, modal frequency, mean, median • box-and-whisker plot, box plot, stem-and-leaf diagram, stem plot • histogram, cumulative frequency, relative frequency, frequency density • random sampling, stratified sampling, bias

UNIT 10A.4 10 hours






10A.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

10A.1.4 Break down complex problems into smaller tasks.

10A.1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.

10A.1.12 Synthesise, present, interpret and criticise mathematical information presented in various mathematical forms.

9.8.1 Identify questions or problems that can be answered or solved by collecting, organising, representing, analysing and interpreting data.

10A.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.

10A.8.2 Locate obvious sources of bias within a sample.

10A.8.3 Know that different types of data can be collected from samples – qualitative/categorical data (e.g. eye colour, male, female) and quantitative data (e.g. age, height, lifespan, mortality rates) – and that quantitative data may be discrete (e.g. number of defective items in a production process) or continuous (e.g. weight); understand the concept of a random variable.

9.8.3 Calculate the mean, range and median of small sets of discrete or continuous data; identify the modal class and estimate the mean, median and range for sets of grouped data.

10A.8.4 Plan surveys and design questionnaires to collect meaningful primary data from representative samples in order to test hypotheses about, or estimate, characteristics of the population as a whole.

11A.15.1 Use a calculator with statistical functions to aid analysis of large data sets, and ICT packages to present statistical tables and graphs.

10A.8.8 Construct histograms, grouping continuous data when necessary. 9.8.4 Construct and interpret frequency diagrams, choosing appropriate equal class intervals.

10A.8.9 Plot cumulative frequency distributions, grouping continuous data when necessary.

1 hour

Data types and data representation

2 hours

Sampling

2 hours

Questionnaire design

2 hours

Histograms

2 hours

Stem-and-leaf diagrams, box-and-whisker plots

1 hour

Cumulative frequency diagrams and their use 10A.8.10 Draw stem-and-leaf diagrams and box-and-whisker plots and use

them in presentations of findings.

Unit 10A.4


Activities


1 hour

Data types and data representation Know that different types of data can be collected from samples – qualitative or categorical data (e.g. eye colour, male, female) and quantitative data (e.g. age, height, lifespan, mortality rates) – and that quantitative data may be discrete (e.g. number of defective items in a production process) or continuous (e.g. weight); understand the concept of a random variable.

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

Representation of data

Class discussion Students will have an understanding of the nature of statistics both from their previous studies and from general knowledge. Draw out their understanding of: • categorical and numerical data; • discrete and continuous numeric data; • grouped numeric data.

Students should know how to represent categorical data (bar charts, pie charts) and be aware of some of the issues of accurate representation of data.

Discuss how to use pie charts to compare data. Explain that a pie chart can show only the relative proportions of data, and does not show the size of the populations. Two pie charts side by side with different radii can be used to represent information (such as life expectancy for men and for women). It soon becomes clear that the areas of the two circles must be in proportion to the size of the populations. This discussion is a useful foundation for the consideration of unequal groups in histograms (where area is again proportional to frequency).

Remind students of the distinction between discrete and continuous data. Ask them to think in terms of lists: it is possible to make a listing of discrete values (shoe sizes, say) but this is not possible with continuous data (mass or height, for example).

Explain that discrete quantities can be measured in a counting process. They change in a discontinuous way. For example, the number of trees in a plantation is a discrete variable over a given period of time. However, the height of a particular tree in the plantation is not a discrete variable – it is continuous. In growing from 360 cm to 361 cm in height, we assume that every possible value between 360 and 361 has been attained. Quantities such as length, mass, temperature, speed and time are continuous variables.

Students will have met grouped data before. Remind them how it arises and why it is necessary to measure data in that way. Remind them also (for the purposes of calculating means and, later, for drawing histograms) of the need to specify the precise class boundaries in use.

For example, if a survey of ages in years is taken for a given set of people, the data will necessarily be grouped: a person of age 25 years has a true age at the time of the sample of somewhere between exactly 25 years and exactly 26 years. This means that in the calculation of average age the representative age for the 25-year-olds will be 25.5 years. Similarly, age groupings such as 25–29 years imply class limits of 25 to 30 years exactly, so a representative mid-interval value would be 27.5 years rather than 27.

On the other hand, if heights are grouped as 25–29 cm, 30–34 cm, 35–39 cm, and so on, the boundaries for the lengths in the first grouping will be 24.5 to 29.5 cm, since we assume that a length given as 25 cm is to the nearest cm and so lies between 24.5 and 25.5 cm.

These ideas are not difficult but they require careful thought.

Example of possible misinterpretation These pie charts show some information about the ages of people in Greece and in Ireland.

• Are there more people under 15 in Greece or in Ireland?


Unit 10A.4



Exercise Give students a selection of questions that cover this ground in a thought-provoking way. Keep any arithmetic as simple as possible to allow students to focus on the issues.

Questions should cover: • a selection of data to be categorised as discrete or continuous; • the calculation of the mean from grouped data.

Sampling

Class discussion Begin with: • why sampling is necessary; • the types of sample which can be drawn.

The two main types of sample are random and stratified. Examples to illustrate these are easily found in a typical school situation. • If some sort of survey is to conducted, it will be time-consuming to ask each student in a

school of, say, 1000 students. The information required could probably be obtained by restricting the survey to a sample only.

• If the survey concerns diet or size of family, for example, which are unlikely to be dependent on age, a random sample would suffice.

• If the survey concerns shoe sizes or some other question for which the response is likely to be age dependent, a stratified sample (where students are stratified by grades, for example) would be much more appropriate.

Encourage students to think out why samples may be biased, and try to tie these to their own instincts and situations. Discuss ways in which possible bias can be avoided.

Stress the need to design a sampling process carefully before going out to collect the 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. If a school has 100 students listed alphabetically, and a sample of 10 is drawn by using a random number between 1 and 10 for the selection of the first one from the list, and every 10th 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).

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

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.

Locate obvious sources of bias within a sample.

Exercises Focus practice questions on design rather than calculation. Present situations which require responses along these lines. • Why is it necessary or appropriate to survey a sample only? • What sort of sampling process is appropriate? • How would the sampling be carried out (precisely)?

Some arithmetic is appropriate in apportioning sample sizes to the strata, for example, but keep this to a minimum.



2 hours

Questionnaire design 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.

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

Plan surveys and design questionnaires to collect meaningful primary data from representative samples in order to test hypotheses about, or estimate, characteristics of the population as a whole.

Designing a questionnaire

Class discussion and project There are several important issues to get across here. They 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 there is a question that will certainly get a response from everyone to whom it is

put.

Ask students to work in groups and to discuss 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 bad 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)

2 hours

Histograms Construct histograms, grouping continuous data when necessary.

Histograms with different class widths

Class discussion Remind students how to draw a histogram with equal class intervals.

Introduce unequal class intervals. Address the question visually. Here is an example of how to do so.

The two data sets on the right are the same grouped frequency distribution, except that in the first case the last two groups have been combined.

0–10 6 0–10 6

10–20 12 10–20 12

20–30 20 20–30 20

30–40 23 30–40 23

40–60 27 40–50 18

50–60 9



The possible histograms are shown on the right. • Students will generally accept the first histogram as ‘correct’ even without the vertical scaling;

they will happily propose the frequency as the appropriate way to determine the scaling used. • The second histogram matches the first on the first four columns, but the combined box looks

quite wrong, although it corresponds to the frequency scaling. • The third figure looks right, but only by virtue of reducing the frequency assigned by a factor

of two.

These ideas motivate the use of frequency density. One or more students may be prepared to suggest that measure as a way of reconciling the diagrams to the figures. It is then a short step to arrive at the conclusion that, on a histogram, frequency is represented not by height but by area, and in consequence:

frequency area width × height

frequencyheightwidth

= =

⇒ =

This motivates the definition of frequency density.

A Java applet that demonstrates the changes of scale involved in using density functions rather than frequencies can be found at Waldomaths (www.waldomaths.com → 16–19 → statistics → histograms)

Show students how to calculate an estimate of the mean and to find a median from a table of frequency densities.

Example The table shows the frequency densities of data for people who swam 1.5 km. Complete the table to show the frequencies. • 304 people took part. Calculate an estimate of the mean time for this event. • Explain why the median time for the event must be between 22 and 27 minutes. • Calculate an estimate of the median time for this event.

Time t

(minutes) Frequency

density Frequency

17 ≤ t < 22 16.0 80

22 ≤ t < 27 28.0

27 ≤ t < 32 12.4

32 ≤ t < 52 1.1

Exercises This needs little follow-up other than practice at using frequency density. Encourage students to mark histograms with an area box with a marked frequency to indicate the scale. This is more readily intelligible than a frequency density scale, even though it is always implicit in the drawing of the figure.



Stem-and-leaf diagrams and box-and-whisker plots

Class exposition These charts are relatively new and are unlikely to have been met by students before this stage. There is nothing particularly difficult about them. Students should readily see their value as visual representations of data.

Remind students of the meaning of median, and interquartile range. Use a spreadsheet such as Microsoft Excel and a graph plotter such as Autograph to demonstrate the processes involved in producing a stem-and-leaf diagram and a box-and-whisker plot.

For more on this see HyperStat (davidmlane.com/hyperstat/A28117.html).

2 hours

Stem-and-leaf diagrams, box-and-whisker plots Draw stem-and-leaf diagrams and box-and-whisker plots and use them in presentations of findings.

Example

Spreadsheet data The table on the right shows the beginning of fifty random numbers between 0 and 50 generated by Excel. In cell A1 enter the formula = INT(50*RAND()) and replicate this down the column.

The stem-and-leaf diagram below is taken from the results box in Autograph after pasting the column in as raw data.

Stem-and-leaf diagram (or stem plot) 0: 0 1 3 5 6 7 7 8 8

10: 0 0 1 1 1 2 3 3 4 7 7

20: 0 0 0 0 2 2 3 4 4 6 7 7 9 9

30: 0 0 0 1 3 4 6

40: 3 3 4 5 7 7 7 8 8

Box-and-whisker plot (or box plot) The box-and-whisker plot on the right is produced by Autograph.

There is also a useful applet Box plot on the website nlvm.usu.edu/en/nav/vlibrary.html.

Box-and-whisker plot

Exercise and project Ask students to: • work through the process of drawing stem-and-leaf diagrams; • draw a box-and-whisker plot.

Check that each student understands the steps involved.

Set up a project or projects so that students collect their own data and use ICT to represent them in tables and box-and-whisker plots.

Make sure that students realise that they have a method for locating data and for comparing the spread.

Example project An example of a project involving collaboration between two schools is the measuring of heights of two groups of students and displaying the results for males and females with separate box-and-whiskers but on the same diagram. Students can email their data to their partner school, or send the data on CD.

This work will generate more interest if there are several different projects under way simultaneously so that students not only gain from the technique in use but appreciate results displayed and explained by others.



Cumulative frequency diagrams

Class discussion Explain that where data are available only as a grouped frequency distribution, there is a need for estimates of spread similar to the length of the box in a box-and-whisker plot. With the data in groups a different approach is needed.

Approach the cumulative frequency graph as an extension of work on the histogram. Almost certainly some of the data considered under that topic can be reused with little further work. Emphasise the connection between histograms and cumulative frequency graphs: specifically, the right-hand class boundaries of the histogram boxes are the values used as abscissas on a cumulative frequency plot.

Opinions differ on whether a cumulative frequency plot should be joined by straight line segments or by a smooth curve. The latter seems more aesthetically pleasing; the former has the advantage that the process can be replicated by calculation so that all should get the same answer.

1 hour

Cumulative frequency diagrams and their use Plot cumulative frequency distributions, grouping continuous data when necessary.

The diagram here was produced from the earlier histogram data by Autograph. The usual technique for drawing and determining the interquartile range is illustrated.

Class exercise Again students should practise the technique in full but should move quickly to using ICT for practical tasks.

Bitesize has a revision page on histograms (www.bbc.co.uk/schools/gcsebitesize/maths/ datahandlingih/scatterdiagramsirev3.shtml).


Assessment


A scientist wanted to investigate the lengths of eggs from a particular breed of hen. Taking a sample of 80 eggs, she measured the length of each one and grouped the data as follows:

Length (l) in cm 4.4 ≤ l < 5.0 5.0 ≤ l < 5.4 5.4 ≤ l < 5.8 5.8 ≤ l < 6.3 6.3 ≤ l < 6.5

Frequency 4 20 36 16 4

Complete a histogram to show this information. Write the frequency density on each part of the histogram.

Calculate the mean length of the eggs in her sample. Discuss how to calculate best estimates for the modal and median values of the lengths of the eggs in the sample.


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

Number of minutes after 1700h

0 ≤ x < 5 5 ≤ x < 10 10 ≤ x < 20 20 ≤ x < 50 50 ≤ x < 60

Number of cars leaving

74 115 248 1174 189

Complete the histogram 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 ≤ x < 5 minutes. Explain what is meant by frequency density with regard to cars leaving the car park.

A student recorded the heights of all the girls in Grade 6 in a school. She summarised her results, then drew the box plot on the right. The student then compared the heights of Grade 6 boys in another school with the Grade 6 girls.

• The shortest boy was the same height as the shortest girl.

• The range of boys’ heights was greater than the range of girls’ heights.

• The interquartile range of boys’ heights was smaller than the interquartile range of girls’ heights.

Draw what the box plot for boys could look like.

There are 120 girls in Grade 8 in the girls’ school. The cumulative frequency diagram on the right shows information about their heights. Compare the heights of Grade 8 girls with the heights of Grade 6 girls.

Unit 10A.4



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.

A school for boys and a school for girls each enters students for the same mathematics examination. The marks are shown on the right.

Draw back-to-back stem-and-leaf diagrams to represent these scores. Compare the performances of the girls and the boys, explaining your methodology and findings.

Using the above data, plot a cumulative frequency graph for the marks of the girls. What was the median score? What was the interquartile range of the distribution of marks? Draw a box-and-whisker plot to represent the girls’ marks.

Draw a relative frequency histogram for these data for both the boys and the girls, explaining how the data were grouped and the meaning of each bar of the histogram.

The girls’ marks were: 97 98 57 45 63 75 87 34 56 28 67 89 45 61 53 49 81 32 23 45 47 72 34 54 23 100 76 47.

The boys’ marks were: 67 87 83 92 34 31 23 25 29 39 89 91 54 47 41 50 77 18 89 10 26 62 39 14 90.

69 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.5 | Geometry and measures 1 © Education Institute 2005

GRADE 10A: Geometry and measures 1

The Earth as a sphere, radian measure and rates

About this unit This is the first of two units on geometry and measures for Grade 10 advanced. It builds on the work on geometry and measures in Grade 9.





Previous learning To meet the expectations of this unit, students should already be able to find the circumference and area of circles and the volume and surface area of right prisms and cylinders and related solids. They should be able to solve simple problems involving average speed or density, using a calculator as appropriate.

Expectations By the end of the unit, students will use radians as a measure of angle, and dimensionally correct units for length, area and volume. They will solve problems involving rates and compound measures. They will use formulae to calculate the length of an arc and the area of a sector of a circle, the area of any triangle, trapezium, parallelogram or quadrilateral with perpendicular diagonals, and the surface area and volume of a right prism, cylinder, cone, sphere and pyramid. They will use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface.

Students who progress further will visualise three-dimensional situations readily by drawing appropriate plane diagrams. They will calculate lengths, areas and volumes of geometrical shapes, and work with compound measures, including density, average speed and acceleration, measures of rate and population density, using appropriate units and dimensions.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector (optional) • computers with Internet access for students • scientific calculators for students • cardboard, scissors and pen for visual aids • football and elastic band • geographical globe

Key vocabulary and technical terms Students should understand, use and spell correctly: • longitude, latitude, meridian, great circle, radius of latitude • speed, density, scale • arc, sector, radian • cone, sphere, cylinder, pyramid

UNIT 10A.5 10 hours



10 hours SUPPORTING STANDARDS Grade 8 and 9 standards



10A.3.6 Understand the multiplicative nature of proportional reasoning; form, simplify and compare ratios, and apply these in a range of problems, including mixtures, map scales and enlargements in one, two or three dimensions.

9.6.1 Find the area of plane shapes related to circles.

9.6.2 Find the volume and surface area of right prisms and cylinders and related solids.

10A.7.1 Find perimeters and areas of rectilinear shapes and volumes of rectilinear solids; find the circumference and area of a circular region, and the surface area and volume of a right prism, cylinder, cone and pyramid, and a sphere, using dimensionally correct units.

11A.11.1 Calculate lengths, areas and volumes of geometrical shapes.

10A.7.2 Use radians to calculate sector areas and arc lengths.

10A.7.3 Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface.

8.7.2 Solve problems involving average speed, distance or time, using a calculator as appropriate.

3 hours

The Earth as a sphere

2 hours

Radian measure

3 hours

Areas and volumes

2 hours

Rates

8.7.4 Know that density = mass/volume; solve problems involving calculating density.

10A.7.4 Solve problems involving compound measures, including average speed, such as cost per litre, kilometres per litre, litres per kilometre, population density (number of people per unit area), density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

11A.11.2 Work with compound measures, including density, average speed and acceleration, measures of rate (such as rate of growth of income), and population density (number of people per unit area), using appropriate units and dimensions.

Unit 10A.5


Activities


3 hours

The Earth as a sphere Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth’s surface.

Using latitude and longitude in calculations

Class discussion Discuss with students what they know about latitude and longitude from their other studies and general knowledge.

Various circles are implicated: circles of latitude and circles of longitude, as well as great circles. Use a cardboard circle folded along a diameter as a visual aid. The concept of the great circle will be least clear, although most students will probably have an intuition that the great circle through two points on a sphere gives the shortest route between them.

To illustrate the great circle and the shortest distance use, say, a plastic football and an elastic band as visual aids. The football should have the equator, a circle of latitude and a circle of longitude marked on it. Stretch the elastic band between two well-spaced points on the circle of latitude; it will spring away from the circle of latitude, demonstrating that this does not give the shortest distance. The idea of a unique great circle linking the two points will be clear to students after seeing the experiment and trying it for themselves.

Explain how to calculate arc length along a circle of latitude and along a circle of longitude. Show how to choose the appropriate circle from a diagram such as figure 1, redrawing it on a piece of paper so that it appears undistorted. The calculation of distance from specified angles and vice versa will appear straightforward.

In figure 2, the radius of the circle of latitude (marked by r) is in the horizontal plane, as is the radius of the equatorial circle (one of the Rs). In this representation of a circle of longitude through N, E, D and S, we have r = R cos x°.

The techniques for students to master are: • calculation of arc length ED given the latitude of E; • calculation of the radius of the circle of latitude centre C given the latitude of E; • calculation of arc length GE given the latitude of the two points and difference in longitude

between them.

Discuss each of these so that students become adept at using and redrawing diagrams.

This topic rapidly becomes difficult. For further study, see the discussion of spherical geometry at Ask Dr Math (mathforum.org/library/drmath/ view/57667.html).

Figure 1

Figure 2

A cardboard circle which represents SAGNEDS when folded along a diameter is a useful visual aid to accompany this diagram.


Exercises These should include examples on: • calculation of arc length given the radius of a circle of latitude; • calculation of the radius of a circle of latitude given the latitude angle; • the reverse of the first two; • combinations of the first two; • revision of bearings so that short-distance problems where a plane calculation will do are

covered.

Take the Earth to be a sphere of radius 6380 km.

In this unit the only great circle arc lengths calculated are those joining points on the same circle of latitude. The general problem of solving a spherical triangle – i.e. a triangle on a sphere, where the sides are arcs of great circles – is beyond the scope of this unit. It is covered in detail on a page of Cambridge University’s site on the history of astronomy at www.hps.cam.ac.uk/starry/sphertrig.html.

Unit 10A.5



Great circle distances

Class discussion Remind students that the shortest distance between two points on the surface of a sphere is along a great circle. Focus again on the case of two well-spaced points on a circle of latitude.

This time use a diagram which shows the great circle. The additional arc joining G to E in figure 3 – compare with figure 1 – shows the arc to be calculated.

Go slowly through the steps needed to solve the problem: • the added arc GE is an arc of a great circle centre O; • to calculate arc GE requires angle GOE, since the radius is known; • to calculate angle GOE requires consideration of the two triangles GOE and GCE, which

have a common side, chord GE (the straight line); • CE can be calculated from angle GCE, which is the difference in latitude.

All the steps are easy, but the visualisation of each triangle and multi-steps make it a considerable challenge. Once students have grasped that chord GE is the connecting link, set about the solution step by step (as above) with sketches of triangles in focus for each step.

2p° is known – it is the difference in latitude – so, in the second triangle, we have: chord GE = 2r sin p°

= 2R cos x° sin p° where x° is the latitude angle, also given.

Similarly, in terms of the unknown angle q°: chord GE = 2R sin q° So 2R cos x° sin p° = 2R sin q° ⇒ q° = arcsin (cos x° sin p°)

Then calculation of the arc length GE is just 2 .360q R×

Figure 3

Support the work with the Great Circle applet at nlvm.usu.edu/en/nav/vlibrary.html.

Class exercise Give students the opportunity to do the calculation just discussed for a number of straightforward cases. Include one case where they must compare the distances of separation resulting from the great circle route and from the circle of latitude route.



2 hours

Radian measure Use radians to calculate sector areas and arc lengths.

Radian measure

Class discussion Introduce radians to students. Include: • the definition of angle measure in radians; • the resulting formula for arc length; • the corresponding one for sector area.

Exercises Include: • routine conversions from degrees to radians and vice versa; • calculation of a radian angle as an intermediate step on the way to calculating sector area.

Take the opportunity to review calculator methods and practice since the material is straightforward.

The real justification for radians lies in the differentiation of trigonometric functions, where radian measure makes the results as simple as possible. The advantage of introducing radian measure at this stage is the easy formula that results for the area of a sector.

It is worth asking students why the same number, π, occurs in both the formula for the circumference of a circle and for the area. If students are not responsive on this point, revise the proof that the area of a unit circle is π.

Areas and volumes of standard solids

Class discussion Ask students to tell you the formulae that they know for the area or volume of shapes. Use the discussion to emphasise the links between different results, such as: • the link between the circumference of a circle and its area formula; • the fact that cylinder and cube are special cases of a right prism; • the fact that the curved surface area of a cylinder is related to the area of a rectangle.

Then present new formulae for: • the volume of a sphere; • the curved surface area of a sphere; • the volume of a pyramid; • the volume of a cone.

The interconnections between these are in some cases intuitively obvious – the occurrence of 1⁄3 in both pyramid and cone volumes is intriguing and the formulae for the sphere again make use of π.

3 hours

Areas and volumes Find perimeters and areas of rectilinear shapes and volumes of rectilinear solids; find the circumference and area of a circular region, and the surface area and volume of a right prism, cylinder, cone and pyramid, and a sphere, using dimensionally correct units.

Understand the multiplicative nature of proportional reasoning; form, simplify and compare ratios, and apply these in a range of problems, including mixtures, map scales and enlargements in one, two or three dimensions.

Exercises These should be extensive. There are new results to master and old techniques to practise, and there are some interesting conceptual applications. Examples to include are: • routine examples of area and volume calculations, using formulae in a straightforward way; • composite figures, such as a hemisphere stuck to a cone by a common circular base; • pouring problems, such as combining contents of two different cylindrical jugs into a third

where the depth is to be determined; • reverse problems, such as determining a sphere’s radius from its volume; • novel ideas, such as calculating the volume of material needed to make a thin balloon (given

its radius and thickness) by multiplying its curved surface area by its thickness.



Practical arithmetic

Class discussion Students have met distance–speed–time calculations in Grade 9. Draw from their existing knowledge answers to questions posed in terms of distance and time to get speed, and with the correct units.

Consider expressions such as:

150 km 1.25 km min40 min

−=

It is clear how the units arise from the way the question is posed. This leads to the problem of unit conversion. We have

1

1

1km1km min1min1km 60 min (since 60 min 1h)1min 1h60 km (since min cancels)

1h60 km h

−

−

=

= × =

=

=

This is a simple idea but needs thorough practice.

Refer to other physical quantities defined, like speed, in terms of others. For example:

mass forcedensity = , pressure = volume area

Units for these quantities arise directly from their definitions, just as with speed. The idea generalises to any form of rate, such as rate of consumption in litre min–1, rate of temperature fall in degree min–1, and so on. In each case, unit conversions can be handled in the same way.

Mention special cases which may be known to students, such as power = work / time, where the special unit 1 W (watt) is exactly the same as 1 J s–1 (joule per second).

2 hours

Rates Understand the multiplicative nature of proportional reasoning; form, simplify and compare ratios, and apply these in a range of problems, including mixtures, map scales and enlargements in one, two or three dimensions.

Solve problems involving compound measures, including average speed, such as cost per litre, kilometres per litre, litres per kilometre, population density (number of people per unit area), density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

Exercises These should bring together the following elements: • unit conversion problems for familiar quantities such as speed and density; • rate calculations where two quantities have to be divided; • comparisons of rates (e.g. km h–1 and m s–1); • use of map scales to find corresponding distances, areas and volumes; • revision of mensuration where areas and volumes are related to time in rate calculations.


Assessment


The volume of a pyramid is (base area × perpendicular height). Calculate the perpendicular height of a pyramid with a square base with side 4 cm and a volume of 48 cm3.

This prism was made from three cuboids.

Show that the area of the cross-section of the prism is 24x2 + 3xy.

The volume of the prism is 3x2(8x + y). What is the depth of the prism?

A round peg, of radius r, just fits into an equilateral triangular hole.

What proportion of the hole is filled by the peg?


A manufacturer makes party hats shaped like hollow cones. To make the hats she cuts pieces of card which are sectors of a circle, radius 24 cm. The angle of the sector is 135°.

a. Show that the arc length of the sector is 18π cm.

b. The sector is joined edge to edge to make a cone. The edges of the sector meet exactly with no overlap. Calculate the vertical height of the completed hat.

A satellite is 1500 km above the Earth. It has a camera with a 50° angle of view with which it surveys the Earth below. Draw a diagram to represent the satellite and its camera in relation to the Earth. Calculate how far apart the two furthest points on the Earth are that can be photographed by the satellite at any one time. Take the Earth to be a sphere of radius 6378 kilometres.

Unit 10A.5



A plane flies from Doha to Karachi almost along the line of latitude 25 degrees north. Doha is at longitude 51 degrees east approximately and Karachi is at longitude 67 degrees east approximately. How far is it from Doha to Karachi along this route?

An oil tanker sails 350 km from Doha towards Dubai on a bearing of 090° and then from Dubai towards Al Kuwayt on a bearing of 310°. Al Kuwayt is about 600 km from Doha. Approximately, how far is it from Dubai to Al Kuwayt?

A satellite passes over both the north and south poles, and it travels 800 km above the surface of the Earth. The satellite takes 100 minutes to complete one orbit.

Assume the Earth is a sphere and that the diameter of the Earth is 12 800 km. Calculate the speed of the satellite, in kilometres per hour.

A water tank is filled through a hosepipe connected to a tap. The rate of flow through the hosepipe can be varied. A tank of capacity 4000 litres fills at a rate of 12.5 litres per minute. How long in hours and minutes does it take to fill the tank?

Another tank takes 5 hours to fill at a different rate of flow. How long would it take to fill this tank if this rate of flow is increased by 100%? How long would it have taken to fill this tank if the rate of flow had been increased by only 50%?

This tank, measuring a by b by c, takes 1 hour 15 minutes to fill.

How long does it take to fill a tank measuring 2a by 2b by 2c at the same rate of flow?



Functions and graphs

About this unit This is the second of three units on algebra for Grade 10 advanced. It builds on Unit 10A.3, Algebra 1.





Previous learning To meet the expectations of this unit, students should already be able to write and solve linear equations. They should be able to use a graphics calculator and a function graph plotter to plot graphs and to find the gradients of lines parallel and perpendicular to a given straight line. They should be able to solve a pair of simultaneous linear equations by elimination and by substitution, and to use graphical methods to find an approximate solution.

Expectations By the end of the unit, students will use proportional reasoning to solve problems involving scale, ratios and percentages. They will use algebraic methods to solve linear equations and a pair of simultaneous linear equations. They will plot and interpret straight line graphs, and graph regions of linear inequality. They will use function notation. Through their study of linear graphs, and the solution of the related equations, students will begin to appreciate numerical and algebraic applications in the real world. They will use ICT to solve problems.

Students who progress further will make ready connections between abstract definitions and their graphical expression. They will easily see the connections between geometrical ideas and their algebraic implications. They will find, graph and use inverse functions and inverse proportional relationships. They will use a graphics calculator, including the trace function, to show approximate solutions to physical problems. They will find exactly by analytical methods and approximately by graphical methods the solution set of two simultaneous equations, one linear and one quadratic.



• dynamic geometry system (DGS) such as: Geometer’s Sketchpad (see www.keypress.com/sketchpad) Cabri Geometrie (see www.chartwellyorke.com/cabri.html)

• computers with Internet access, graph plotting and dynamic geometry software for students

• graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • relation, function, gradient, region, co-domain, image, injection, surjection,

bijection, inequality, inequation

UNIT 10A.6 14 hours






9.3.6 Write and solve linear equations, including simple cases of fractional linear equations, and apply these skills to solving problems; verify the solution.

10A.1.12 Synthesise, present, interpret and criticise mathematical information presented in various mathematical forms.

9.4.1 Use a graphics calculator and a function graph plotter to plot graphs.

10A.5.1 Investigate a range of mathematical and physical situations to develop the concepts of function, domain and range, recognising one-to-one and many-to-one mappings as functions and a one-to-many mapping as a relation but not a function.

11A.5.13

9.4.2 Find the gradients of lines given by y = mx + c; understand the idea of slope; find the gradients of lines parallel and perpendicular to y = mx + c.

10A.5.2 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

Find, graph and use the inverse function of those functions 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.

4 hours


2 hours

Simple proportion

4 hours

Straight line graphs

4 hours

Intersecting straight lines and regions

9.4.3 10A.5.3 Plot a graph to show the relationship between two variables given quantitative information between the variables in tabular or algebraic form.

Use graphical methods to find the approximate solution of a pair of simultaneous linear equations with two unknowns, on paper and using ICT.

10A.5.4 Use a graphics calculator or graph plotter and pencil-and-paper methods to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts.

10A.5.5 Recognise when a graph represents a functional relationship between two variables and when it does not.

10A.5.6 Translate the statement y is proportional to x into the symbolism y ∝ x and into the equation y = kx, and know that the graph of this equation is a straight line through the origin and that the constant of proportionality, k, is the gradient of this line.

11A.5.10

10A.5.7 Know that if two coordinate variables are connected by a straight line graph that passes through the origin, then each coordinate variable is proportional to the other; use relevant information to find k.

Understand the statement y is inversely proportional to x and set up the corresponding equation y = k / x; know some characteristics, including that x ≠ 0 and that x = 0 is an asymptote to the curve, as is y = 0; study examples of inverse proportionality.

10A.5.8 Identify common examples of two linear quantities varying in direct proportion to each other.

Unit 10A.6





10A.5.9 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.

10A.5.10 Plot the graphs of the equations in 10A.5.9; know the meanings of gradient of the line and intercept on the x- or y-axis and relate these to the coefficients a, b and d, or to the coefficients m and c.

10A.5.11 Construct the Cartesian equation of a straight line from its graph alone, or from the knowledge of the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line.

10A.5.12 Know the condition for two straight lines to be parallel or perpendicular, including the special cases of one of the lines being parallel to either axis.

9.3.7 Write and solve simultaneous linear equations with two unknowns by elimination and by substitution, and apply these skills to solving problems; verify the solution.

10A.5.13 Read off the coordinates of the point of intersection, given the graphs of two intersecting straight lines; find exactly, by algebraic means, the coordinates of the point of intersection of two lines, given their equations.

11A.5.11 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.

10A.5.14 Interpret the solution set of the simultaneous equations E1 and E2, where E1 and E2 are the equations of two straight lines.

11A.5.8 Find exactly by analytical methods and approximately by graphical methods the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x.

10A.5.17 Identify some other common examples of proportional variation.

10A.5.18 Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions.


Activities



Functions and relations

Class discussion Remind students that a graph defines a relationship between two quantities, usually marked off along the horizontal and vertical axes.

Explain that a function is a rule which processes an input or inputs to produce an output. Use an example of a set of instructions for making a bookshelf: the inputs are the raw materials – wood, screws, brackets – and the output is the finished bookshelf. A set of instructions is needed to transform one into the other. In this analogy, the role of function is taken by the set of instructions.

Give a more mathematical example. Consider the rule that maps an integer onto its final digit. This is a good example because it allows discussion of the concepts of domain and range. In this case the two are distinct; one is infinite and the other finite.

Consider a second example and ask for the features which make them different: for example, the function which maps each integer x onto 2x. A further example (that broadens discussion from functions to relations) is shown in the diagram on the right. Point out the non-uniqueness of the image.

Draw out the set of definitions: • domain – the set of elements on which the function is defined; • range – the set of image elements (usually a set of numbers); • relation – a connection between the members of a set, or between the members of two sets one to

another (the domain and the range); • many-to-one – a relation in which at least one element of the range corresponds to more than one

domain value; • one-to-one – a relation in which each range element corresponds to a unique domain element; • function – a relation which is many-to-one or one-to-one, i.e. the image element is unique for each

domain element.

Stress that this way of looking at the situation leads naturally to the idea of a function as a set of paired values. Identify what is thought of intuitively as a ‘rule’ with a set of ordered pairs; this is equivalent to thinking of a function on the one hand as a formula and on the other as a graph.

The different possible notations: • f(x) = x2 + 3x – 5 • f: x → x2 + 3x – 5 should be covered and used.

For more able students, other standard terms can also be introduced, such as co-domain, image, injection, surjection, bijection.

4 hours


Investigate a range of mathematical and physical situations to develop the concepts of function, domain and range, recognising one-to-one and many-to-one mappings as functions and a one-to-many mapping as a relation but not a function.

Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

Plot a graph to show the relationship between two variables given quantitative information between the variables in tabular or algebraic form.

Use a graphics calculator or graph plotter and pencil-and-paper methods to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts.

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

Exercises

Set exercises which cover: • specifying (or listing – for finite sets) the elements of domains and/or ranges;

• introducing functions, such as f(x) = 11x −

or f(x) = 2x + or f(x) = sin xx

, where the domains are

necessarily restricted.

The idea of inverse function can be discussed if this seems natural.

Note sin xx

requires radian measure.

Unit 10A.6



Discussion

Get students to think how and why greater precision of definition is necessary in particular cases.

Examples

f(x) = 11x −

The graph makes clear that x = 0 cannot be part of the domain of definition. Students should expect to anticipate this sort of difficulty as well as to interpret it from the graph produced by Autograph or on a graphics calculator.

f(x) = 2x +

In this case, x < –2 is excluded from the domain of definition.

Note that applications such as Autograph will produce a graph even when the input is defective. This requires critical use by students, although the necessary caveats will usually be obvious.

Problem

The graph on the right is of the function f(x) = sin xx

.

• What value of x must be excluded from the definition of the function, and why? • What value can you suggest for the function when x = 0 that would make the function

continuous for all values of x?

Functions and their graphs

Class discussion Develop the definition of a function as a set of ordered pairs, one drawn from each of the domain and range. The idea of a set of paired values leads naturally to the idea of a graph; what has been defined is a functional relationship, and the domain and range are the familiar x- and y-axes.

Characterise the distinction between a function and a relation in terms of the appearance of their graphs. Examples to distinguish the two cases are: • y = x2 (a function); • x2 + y2 = 4 (a relation only).

Once students have been taught Unit 10A.12, the connection can be developed fully.



Explain that a vertical line on a conventional set of Cartesian axes must cut a function’s graph only once.

Examples

y = x2

This graph represents a function.

x2 + y2 = 4

This graph represents a relation.

y = √(4 – x2), –2 ≤ x ≤ 2

This graph represents a function.

Exercises Include in the exercises all the types of curves and straight lines considered so far so that they can be reviewed in the light of the function/relation distinction.

Basic ideas of proportion

Class discussion In studying proportion, it is important that the formal algebra and graphical representation stay in step. Keep this parallel development in mind by presenting a number of familiar situations where data can be easily generated (the one exact, the other approximate) and shown on a graph. Examples are: • currency conversions; • a physical experiment to show Hooke’s law.

Ask students to consider two questions: • What are the characteristics of such graphs? (they are straight lines which pass through the origin) • What arithmetic relation is evident in the tables of values? (the ratio of corresponding pairs is

constant)

Students will readily see these as connected, and the way is opened up to define gradient. Do so now if it seems appropriate, although it appears explicitly later in the unit too.

Make students familiar with the formal terminology: if y ∝ x, then y = kx.

The constant which arises in such contexts can often be given a physical meaning, such as density.

2 hours

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

Plot a graph to show the relationship between two variables given quantitative information between the variables in tabular or algebraic form.

Translate the statement y is proportional to x into the symbolism y ∝ x and into the equation y = kx, and know that the graph of this equation is a straight line through the origin and that the constant of proportionality, k, is the gradient of this line.

Identify common examples of two linear quantities varying in direct proportion to each other.

Identify some other common examples of proportional variation.

Exercises Include: • some elementary completion of tables of proportion with some elements missing; • some examples where the constant has to be determined from information given; • some examples of graphs which are and are not examples of proportion.

The idea of the constant as the gradient of such a graph need not be explicit but will prepare for the formal study of the straight line.



Calculator work Ask students to obtain graphics calculator displays of some given straight line graphs and in some cases to read off values from such graphs.

Equations of straight lines

Investigation Get students to use their graphics calculators to investigate straight lines in a way that makes some of their properties immediately obvious. Without further introduction, students can investigate the properties of the equation y = mx + c, and the work can be easily formalised subsequently.

Suitable starting points for investigation are these. • Use your graphics calculator (or Autograph) to investigate the equation y = mx + c. For a fixed value of c of your choice, investigate what happens when you graph the equation for

different values of m. For a fixed value of m of your choice, investigate what happens when you graph the equation for

different values of c. Repeat these exercises until you feel that you can summarise in words the contribution m and c

make to the graph that you get. What properties do the graphs have in all cases?

• Use your graphics calculator (or Autograph) to arrange graphs like the figure on the right, where the rhombus has its vertices at (–1, 0), (0, 2), (1, 0), (0, –2). Write down the equations of the graphs that you use, and account for any patterns.

A resource on straight line graphs is available on MathsNet (www.mathsnet.net/graphs/lines.html).

Class discussion Students should be able to volunteer a number of observations as a result of their investigations. Draw out the following conclusions: • parallel lines have the same value of m, and y = mx + c is parallel to y = mx; • lines with the same value of c have the same intercept c on the y-axis; • (perhaps) perpendicular lines have their gradients related by m1m2 = –1.

Formalise these results in the usual definition of gradient and intercept, with consideration of the case of gradient undefined. The alternative forms ax + by + c = 0 should be related. The rule for perpendicular gradients should also be proved.

If students spot the rule for perpendiculars, this conjecture can lead to the question of how the result can be proved.

Spell out that the case b = 0 does not correspond to any value of m.

4 hours

Straight line graphs

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.

Plot the graphs of the equations shown above; know the meanings of gradient of the line and intercept on the x- or y-axis and relate these to the coefficients a, b and d, or to the coefficients m and c. Construct the Cartesian equation of a straight line from its graph alone, or from the knowledge of the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line.

Know the condition for two straight lines to be parallel or perpendicular, including the special cases of one of the lines being parallel to either axis.

Exercises Include: • finding gradients for pairs of points; • finding the equation of a line through two points; • given a line and a point not on it, find the equation of a parallel line through the point; • given a line and a point, find the equation of a perpendicular line through the point.

This set of exercises can finish with the question of determining where two lines cross as a preparation for the next step.



Investigation Provide some unlabelled printouts of straight line graphs and ask students to reproduce them on their calculators. • Find the equations of these straight line graphs. • Find some more straight line graphs which pass through the point (4, 6).

Finding intersections

Class discussion Remind students that an approximate solution to a pair of simultaneous equations can be obtained by representing them by two lines that intersect. The idea follows on naturally from the coordinate geometry of the straight line.

The graph illustrates the solution of:

2 22 3 6x yx y

− =+ =

Revise the essential ideas.

Simultaneous equations feature in an applet on the website Waldomaths (www.waldomaths.com →14–16 → simultaneous equations).

Take care in developing class exercises to make sure that the arithmetic does not become too intimidating.

4 hours

Intersecting straight lines and regions

Read off the coordinates of the point of intersection, given the graphs of two intersecting straight lines; find exactly, by algebraic means, the coordinates of the point of intersection of two lines, given their equations.

Interpret the solution set of the simultaneous equations E1 and E2, where E1 and E2 are the equations of two straight lines.

Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions.

Exercises Give students some practice in finding the approximate solution of a pair of simultaneous equations.

Problems Extend the work on coordinate geometry to incorporate the new technique. Make sure that students can form the equation of a line joining two given coordinate points.

There are some interesting cases to discover, such as the common intersection of the medians of a triangle and similarly of its altitudes. • Show that for triangle OAB, where A is the point (0, 9) and B (18, 0), the three medians intersect at

a common point (i.e. they are concurrent). Do you think that this result generalises?



Where possible, explore the results using a dynamic geometry system such as Geometer’s Sketchpad or Cabri. For example, construct a triangle with the perpendicular bisectors of its three sides. Draw the circumcircle. Observe the effect of dragging the vertices of the triangle, particularly noting the position of the centre of the circumcircle.

Finding these results in a particular case may motivate a general conjecture and search for proof by pure geometry.

Graphing inequations

Class discussion Many of the diagrams sketched or constructed in the course of this work show the coordinate plane divided into regions. It has not yet been made explicit that the line ax + by = c divides the coordinate plane into two regions: on one side ax + by > c and on the other ax + by < c. Get students to discover this by: • drawing the line ax + by = c; • plotting the value of ax + by at several points.

Allow students to investigate this, using graph paper and plotted points, or using a graphics calculator. They will rapidly discover that the same value is produced in a line of points and that the line is parallel to the original.

Introduce the idea of defining regions by a set of inequations. Give students some formal exercises on this topic using a graphics calculator (or Autograph) and squared paper in tandem; Autograph is particularly good for preparing answers for use in class.

Establish a convention and stick to it. For example: • shade the undesired region so the

‘answer’ remains unshaded when all lines have been drawn;

• use dotted lines for strict inequations (i.e. where the line of equality is not included) and full lines otherwise.

Teach students to: • draw the appropriate line first; • avoid guesswork by testing the origin’s

coordinates in the given inequation to determine which side is required.

Exercises

These should include: • practice at drawing regions defined by a set of inequations; • prescribing equations which define a region displayed in a sketch; • some applications as cases of problem solving; • extension to simple curves.

Problem • What set of inequations defines the clear region in the diagram on the right?


Assessment


If p is a person, state with reasons whether each of the following maps is a function: a. p maps to the place of birth of p; b. p maps to brother of p; c. p maps to nationality of p; d. p maps to teacher of p; e. p maps to mother of p.

The Int x function, written as [x], maps x to the greatest integer less than or equal to x. Find [5.9], [6] and [–4.7].

Ahmed does a parachute jump. He jumps out of the plane and falls faster and faster towards the ground. After a few seconds his parachute opens. He slows down and then falls to the ground at a steady speed.

Which of these graphs shows Ahmed’s parachute jump? Explain why each of the other graphs is wrong.


A rectangular enclosure has a wall on one side and the other three sides are made of metal fencing. The side parallel to the wall has length d metres. The enclosure has an area of 600 m2. Show that the total length, L metres, of fencing is given by L = d + 1200/d. Plot this function using a graphics calculator. Find from the graph the value of d that makes L as small as possible.

A triangle has vertices at the points (1, 1), (5, –4) and (–3, 2). Find the equation of each side.

Solve the simultaneous equations y = 2x – 6 y + 3x = 5 to find the point of intersection of these two lines.

Here are the equations of some straight lines: y = 2x – 7; y = 7 – 2x; y = 2x + 9; y = 14 – 4x; y = –10; y = –10 + 2x; x = 1; y = –0.5x + 8.

List all the pairs of lines that are: a. parallel to each other; b. perpendicular to each other; c. different representations of the same line. From the lists, find pairs of lines that intersect in a unique point and find the intersection point in each case.

Unit 10A.6


GRADE 10A: Geometry 2

Tangent theorems, constructions and loci

About this unit This is the second of two units on geometry for Grade 10 advanced. It builds on the work in geometry in Grade 9 and Unit 10A.2, Geometry 1.





Previous learning To meet the expectations of this unit, students should already be able to deduce properties in a given plane figure using their knowledge of angles and properties of 2-D shapes, and to determine whether two triangles are congruent. They should be able to develop a simple proof, and explain and justify the steps taken to solve a problem.

Expectations By the end of the unit, students will use their knowledge of geometry to solve practical and theoretical problems relating to shape and space. They will prove that the perpendicular from the centre of a circle to a chord bisects the chord and that the two tangents from an external point to a circle are of equal length. They will carry out straight edge and compass constructions, and determine the locus of an object moving according to a rule. They will use ICT to explore geometrical relationships.

Students who progress further will appreciate the connections between practical and intuitive solutions on the one hand and the need for proof on the other. They will be able to prove a wider range of standard circle theorems.

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



• sharp pencil, protractor and ruler for each student

Key vocabulary and technical terms Students should understand, use and spell correctly: • centre, radius, diameter, circumference • arc length, sector, segment, chord • tangent, tangent kite • circumcentre, orthocentre, in-centre, centroid • theorem, converse, corollary, proof by contradiction

UNIT 10A.7 10 hours






10A.1.6 Develop short chains of logical reasoning, using correct mathematical notation and terms.

9.1.4 Explain and justify the steps taken to solve a problem or arrive at a conclusion, orally and in writing.

10A.1.7 Explain their reasoning, both orally and in writing.

9.1.6 Develop a simple proof. 10A.1.8 Generate simple mathematical proofs, and identify exceptional cases.

9.5.1 Use knowledge of angles and properties of 2-D shapes to conjecture or deduce properties in a given plane figure.

10A.6.9 Prove the circle theorems: • The perpendicular from the centre of the circle to a chord

bisects the chord. • The two tangents from an external point to a circle are of

equal length.

11A.8.9

9.5.4 Identify congruent triangles and their corresponding angles and sides; know the conditions of congruence and determine whether two triangles are congruent.

10A.6.10 Perform and justify straight edge and compass constructions, including those to bisect a line, to construct an equilateral triangle with a side of given length, to drop a perpendicular from a point to a line, and to bisect an angle.

10A.6.11 Determine the locus of an object moving according to a rule, including those arising in simple physical situations.

10A.6.12 Investigate Islamic patterns and describe their features.

Prove the circle theorems: • The angle subtended by an arc at the centre

of the circle is twice the angle subtended by the arc at a point on the circle, including, as a special case, the angle in semicircle is a right angle.

• Angles in the same segment subtended by a chord are equal.

• The angle subtended by a chord at the centre of a circle is twice the angle between the chord and the tangent to the circle at an end point of the chord.

• When two chords BC and DE in a circle intersect at A then AB × AC = AD × DE.

• Opposite angles of a cyclic quadrilateral are supplementary.

2 hours

Symmetry in a circle

3 hours

Constructions

3 hours

Tangents and the tangent kite

2 hours

Locus and geometrical patterns

10A.6.13 Use ICT to explore geometrical relationships.

Unit 10A.7


Activities


Using symmetry

Class discussion Remind students that establishing congruence is often an important stepping stone in a geometrical proof; when stuck, it is always worth asking which triangles are congruent.

Students should be thoroughly familiar with the various properties of the isosceles triangle, established in the previous unit on geometry. Rehearse those first. They establish clearly that an isosceles triangle has an axis of symmetry – a geometrical fact on which all the other properties depend.


2 hours

Symmetry in a circle

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

Explain their reasoning, both orally and in writing.

Generate simple mathematical proofs, and identify exceptional cases.

Prove the circle theorem: • The perpendicular from the

centre of the circle to a chord bisects the chord.

Establish the results that: • in triangle ABC, if AB = BC, then p = q; • in triangle HKL, if x1 = x2, then HK = KL. These two results are an early example of a theorem and its converse.

The proofs proceed by constructing the perpendicular from B and K to each base and establishing that the two new triangles are congruent in each case.

A corollary of each result is the symmetrical properties of the original triangle: that all corresponding measurements are equal in the pair of congruent triangles.

Move on to consider the implications of symmetry in a circle.

In the figure on the right, ask students what they know about the circle and the triangle OAB. Expect them – perhaps with prompting – to volunteer that: • OAB is isosceles; • OM should be drawn (because every isosceles triangle has an axis of symmetry).

Use this to define the distance of a chord from the centre of a circle: that distance is OM, the length of the perpendicular from the circle centre to the chord.

Exercises Base these examples on the last figure. They involve practice of: • Pythagoras’ theorem; • elementary trigonometry.

Include plenty of questions that: • give the radius and the length of a chord and require calculation of its distance from the

centre; • give the radius and the distance of a chord and require calculation of its length; • specify a chord by either its length or its distance and require calculation of the angle

subtended at the centre.

Unit 10A.7



3 hours

Constructions Perform and justify straight edge and compass constructions, including those to bisect a line, to construct an equilateral triangle with a side of given length, to drop a perpendicular from a point to a line, and to bisect an angle.

Use ICT to explore geometrical relationships.

Straight-edge and compasses constructions

Class demonstration and discussion Use a dynamic geometry system (DGS) such as Geometer’s Sketchpad to demonstrate the basic constructions quickly: • 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. The discussion will connect easily with the question of justification. Justify everything fully, using the work on isosceles triangles.

Emphasise the need for: • clear diagrams; • lines which are always too long so that they go right through intersection points; • making figures large enough; • avoiding equilateral or isosceles triangles unless explicitly required.

There is an online course in geometrical construction at MathsNet (www.mathsnet.net/campus/construction/ index.html).

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

Exercise Get students to practise their construction skills. 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 the three sides; • discover their 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.

Challenge more able students to construct the circumcentre (C), the centroid (G), and the orthocentre (H) in a single triangle.

Demonstrate that the three points are collinear (defining the Euler line) and also that HG : GC = 2 : 1. This can be demonstrated effectively in DGS or in Lines in geometry, but presents a considerable challenge for proof.



3 hours

Tangents and the tangent kite

Prove the circle theorem: • The two tangents from an

external point to a circle are of equal length.

Use ICT to explore geometrical relationships.

Lines which touch circles

Investigation Use DGS to explore the properties of the tangent to a circle. Begin by discussing what students think a tangent is. Once you have established the idea of a line that touches a circle at one point, set up the investigation as follows.

Part A • Use DGS to draw a circle in the centre of your worksheet; choose a point outside the circle; construct a line through your chosen point to touch the circle – you will need to use eye and

judgement to achieve this.

• Why is it difficult to decide just which point is the point of contact?

• Choose the point you think is at least close to the point of contact – that point has to be on at least one of the line or circle;

join that chosen point to the centre of the circle by a radius; measure the angle between that radius and the tangent.

• Repeat this whole procedure and compare your answers.

• Given that it is unlikely that you have found an exact point of contact between tangent and circle, what do your answers suggest the ideal answer may be?

Allow students to discuss their conclusions freely so that they can convince each other that the angle should be a right angle. They may not be convinced: you may require supplementary discussion to turn a plausible conjecture into a firm assumption.

The figure shows DGS in use for the investigation. The angle measurement is close to 90°. In practice the exact value of 90° is unattainable by this means.

If the concept of tangent is completely new, approach the idea by considering a line that cuts a circle (in two points, obviously) and progressively slides off the circle. You can do that with DGS. As the line slides off the circle: • the two points of intersection come closer together; • the conjecture arises that there is a limiting case where they coincide.

Students may realise that this is like the conjecture – then an axiom – that parallel lines exist. There is no guarantee that such a tangent exists, as the limitation of DGS makes clear; we simply assume that it does.

Prove that the tangent is at right angles to the radius at the point of contact like this.

If the tangent does not meet the radius at right angles, suppose that p < 90° and q > 90°.

Construct the perpendicular from O, the centre, to the tangent, so that it meets it at U. U must be outside the circle, since for U to be on the circle contradicts the assumption of a unique point of contact. Then OU is greater than OT, the radius; but by Pythagoras the hypotenuse must be the longest side: this is a contradiction.

This is a useful example of reductio ad absurdum or proof by contradiction.



Part B In this investigation, you should assume that a tangent may be drawn from a point outside a circle to meet it at one contact point, and that the angle between the tangent and radius at the point of contact is a right angle.

• Use DGS to draw a circle and an external point; draw by trial as many tangents as you can from that external point to the circle.

• How many tangents can you draw?

• For each tangent, draw a line from the centre of the circle perpendicular to the tangent.

• Why must the intersection point be the point of contact with the circle?

Again, give students time to compare their conclusions. Some students may need convincing by argument that there are only two tangents from an external point.

Aim to keep a balance between: • the use of DGS and discussion to develop intuition and hypotheses; • a fully rigorous establishment of results.

Part C • Use DGS to draw a circle, centre O. Mark an external point, T.

Draw the two tangents by eye and construct the radii at their points of contact, A and B, as perpendiculars from the centre.

• Measure TA and TB. What do you notice?

• Measure ∠ATO and ∠BTO. What do you notice?

• Repeat in another case. Do your results generalise?

• Can you prove your conclusions (with pencil and paper, by use of congruent triangles)?

Use the results of this investigation to create general discussion.

Students need time to: • distinguish conjectures from established facts; • appreciate the need for proof to parallel their investigative work; • set up a proof correctly.

In this section students must be able to appreciate the steps by means of which investigation leads to conjecture, and conjecture leads to proposition and proof.

Exercises Focus these – as with the previous work on chords in a circle – with applications that exploit right angles and require use of Pythagoras’ theorem and trigonometry. These will help to reinforce retention of the new properties and key facts as well as the skills of trigonometry. For example: • If ST = 24 cm and OS = 7 cm, find OT. • Prove that SVO is similar to TSO. • Find SV. • Find the area of sector SOU.



Problem Two intersecting circles have a common chord AB. Point C moves on the circumference of the first circle. The straight lines CA and CB are extended to meet the second circle at E and F respectively. As point C moves, what do you notice about the chord EF? Give a proof of your conjecture.

See www.nrich.maths.org for an applet to explore the problem (called Chords, posted September 1998) and a solution.

2 hours

Locus and geometrical patterns

Determine the locus of an object moving according to a rule, including those arising in simple physical situations.

Investigate Islamic patterns and describe their features.

Locus

Class discussion and demonstration This is students’ first introduction to the idea of a locus. There are many practical locus problems which are easy to visualise but which lead to interesting mathematical diagrams.

Begin with the idea of a locus in simple cases, such as: • a point that is a constant distance from a given point (two- and three-dimensional cases); • a point that is a constant distance from a line (clarity of what is meant by distance is

needed); • combinations, such as a point that is a constant distance from the perimeter of a rectangle.

Elaborate these into elementary problems such as the locus of an animal tethered to a point or a line within an enclosure, and related problems.

DGS can be helpful in demonstrating the ideas, particularly if animation is used. The necessary diagrams are quite difficult to set up, however. It is worth constructing a resource for this topic that may then be used again and again.

Simple locus problems In the scale drawing, the shaded area represents a lawn. There is a wire fence all around the lawn. The shortest distance from the fence to the edge of the lawn is always 6 m. On the diagram, draw accurately the position of the fence.

The plan shows the position of three towns, A, B and C, each marked with a ×. The scale of the plan is 1 cm to 10 km.

The towns need a new radio mast.

The new radio mast must be nearer to A than to C, and less than 45 km from B.

Show on the plan the region where the new radio mast can be placed.



Further discussion Use the ‘trace’ facility in DGS to explore the path of a point in different conditions. The diagram is constructed as a circle with a radius drawn in. The point where the radius meets the circle is labelled A, and there is a point B that can be moved up and down the radius. The point B is set so that its path is plotted on the screen using the ‘trace’ feature. For example, using the diagram, say to students: • Describe the locus of point B if I drag point A round the circle. (the locus of B is a circle) • Describe the locus of point B if I drag point A around the circle and decrease the radius of

the circle at the same time. (the locus is a spiral)

Ask students to use a pencil and a length of string fixed at each end to trace out an ellipse. • Why is an ellipse formed? • What properties do you think that an ellipse has?

Ask students to rule a line on a piece of plain paper. Then ask them to choose a triangle, square or rectangle shape and mark one corner of it. They are going to roll, or topple, the shape along the line on the paper and mark the path of the marked corner on the paper. Before they do this, ask them to predict and sketch what they think the locus of the point will be.

The activity can be repeated with different shapes and by using different corners of non-regular shapes to compare the patterns made.

Rolling a triangle round a triangle

Exercises Combine the requirement that students turn precise directions into good and effective diagrams, together with the discovery of interesting new situations. For example: • Draw the locus of a railway wheel’s centre as a train moves along a straight, level track. • Draw the locus of a bicycle wheel’s hub as the bicycle moves horizontally in the same

vertical plane and mounts a stone kerb. • Draw the locus of the end of a piece of string which unwinds from a square-sectioned stick. • Draw the locus of a point on the edge of a railway wheel which begins in contact with the

surface of the rail – not the edge of the flange.

Include in the exercises revision of the calculation of areas (in terrain-covering problems, for example) of geometrical shapes and of lengths. • Find the area swept out by the string that unwinds from the square-sectioned stick

(assuming that it starts taut with the string at the corner of a 4 cm square stick and makes one complete turn of unwinding).

• Find the length of the bicycle wheel hub’s locus if the wheel has radius 30 cm, starts with its point of contact 1 metre from the kerb, and continues until it is 1 metre beyond the kerb edge, and the kerb itself has height 12 cm.

The examples of points at various locations on a railway wheel are fascinating to consider (and difficult to construct with DGS).

The bicycle wheel and the kerb

The beginning of the unwinding string problem



Investigation Conclude with an investigative look at Islamic patterns, where the symmetries and locus problems link closely with the other ideas of this unit. • This pattern is from a mosque in Isfahan, in Iran.

Use this and other Islamic patterns to discuss key features of the pattern (its construction,

reflection symmetries, translations, and so on).

Work on Islamic patterns is available at the site Geometry and Islam (www.askasia.org/frclasrm/ lessplan/l000030.htm).

There are other areas on the same site, accessible from the menus on the homepage at www.askasia.org.

Another useful site for work on Islamic patterns is www.cgl.uwaterloo.ca/~csk/washington/taprats.


Assessment


Construct a square. You may use a straight edge, a pencil and a pair of compasses only.

Use the construction to bisect an angle several times over to construct an angle of 22.5°.

Explain why the construction to bisect an angle works.

A goat is on a rope attached at one corner of a rectangular enclosure. The enclosure measures 10 m by 4 m. The rope is 6 m long. Draw a scale drawing of the enclosure, and shade in the locus in which the goat can move.

Find the locus of all points 3 cm from a circle of radius 5 cm. Discuss how the locus is changed if three dimensions are allowed.

The diagram shows two concentric circles.

AB is a chord of the larger circle and a tangent to the smaller circle.

Given AB = 12 cm, calculate the shaded area, leaving your answer as a multiple of π.

MEI


A tangent to a circle centre C from an external point A has length 15 cm and the circle has radius 8 cm.

Find the length AC.

If the two tangents from A touch the circle at S and T, find the length ST.

Unit 10A.7

97 | Qatar mathematics scheme of work | Grade 10 advanced | Unit 10A.8 | Number and algebra 1 © Education Institute 2005

GRADE 10A: Number and algebra 1

Sequences

About this unit This unit is the first of two units on number and algebra for Grade 10 advanced. It builds on Unit 10A.1, Number, and Units 10A.3 and 10A.6, Algebra 1 and 2.





Previous learning To meet the expectations of this unit, students should already be able to use a graphics calculator to generate sequences and plot graphs. They should be able to extend and find missing terms in sequences, and to use symbols to generalise the relationship between one term and the next, or to describe the nth term.

Expectations By the end of the unit, students will use mathematics to model and predict the outcomes of real-world applications, and will synthesise and interpret mathematical information in various forms. They will sum arithmetic and geometric sequences and investigate the growth of patterns, including in Pascal’s triangle. They will generalise relationships wherever possible and will recognise that even and odd numbers can be written in the form 2m and 2m + 1, where m is an integer. They will convert recurring decimals to exact fractions. They will recognise when to use ICT and when not to, and will use it efficiently.

Students who progress further will make confident use of formulae for arithmetic and geometric series, and have no difficulty with index notation in the latter.

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

Key vocabulary and technical terms Students should understand, use and spell correctly: • words related to sequences: convergent, divergent, oscillatory,

arithmetic, geometric, progression, common difference, common ratio, sum to n terms, Fibonacci sequence, Pascal’s triangle

UNIT 10A.8 12 hours






10A.1.2 Use mathematics to model and predict the outcomes of real-world applications.

10A.1.9 Generalise whenever possible.

10A.1.12 Synthesise, present, interpret and criticise mathematical information

presented in various mathematical forms.

8.5.1 10.1.14 Recognise when to use ICT and when not to, and use it efficiently. Use a graphics calculator to generate sequences and plot graphs.

10A.2.5 Know from definitions that every even number can be written in the form 2m, where m is an integer, and that every odd number can be written in the form 2n + 1, where n is an integer.

8.5.2 Extend and find missing terms in numeric, algebraic or geometric patterns or sequences (term-to-term or position-to-term rules).

10A.4.2 Generate sequences from term-to-term and position-to-term definitions; investigate the growth of simple patterns, generalising algebraic relationships to model the behaviour of the patterns.

11A.4.6 Understand and use formulae for the sum of: • the squares of the first n positive

integers; • the cubes of the first n positive

integers. 8.5.3 Generalise the relationship between

one term of a sequence and the next, or describe the nth term, using symbols.

10A.4.3 Identify and sum arithmetic sequences, including the first n consecutive positive integers, and give a ‘geometric proof’ for the formulae for these sums.

11A.4.3 Recognise an arithmetic progression (AP); sum an arithmetic series and know the formula for the rth term of the series in terms of the first term and the common difference between terms.

10A.4.4 Identify and sum geometric sequences and know the conditions under

which an infinite geometric series can be summed. 11A.4.4

10A.4.5 Convert any recurring decimal to an exact fraction.

Recognise a geometric progression (GP); generate term-to-term and position-to term definitions for the terms of a GP in terms of the common ratio between terms; sum a finite geometric series.

2.5 hours

Working with sequences

2.5 hours

Arithmetic sequences

5 hours

Geometric sequences

2 hours

Pascal’s triangle

10A.4.6 Identify number patterns contained in Pascal’s triangle.

Unit 10A.8


Activities


Sequences

Class discussion Introduce the idea of a sequence of numbers in as varied a way as possible. Invite suggestions as to how the next term can be determined; if the series are sufficiently varied there are obvious cases where the nth term is easily expressed in terms of n, and others where the immediately previous terms need to be known. Establish a notation such as un for a sequence so that the notation implicitly suggests the nth term’s connection with the integer n. Examples should include: • cases of arithmetic and geometric sequences; • squares and cubes; • Fibonacci sequences; • the triangular numbers.

Use the applet Fibonacci at nlvm.usu.edu/en/nav/vlibrary.html to demonstrate and explore the sequence.


2.5 hours

Working with sequences Know from definitions that every even number can be written in the form 2m, where m is an integer, and that every odd number can be written in the form 2n + 1, where n is an integer.


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

Generate sequences from term-to-term and position-to-term definitions; investigate the growth of simple patterns, generalising algebraic relationships to model the behaviour of the patterns.

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

Investigation This work is inevitably investigative, since there are several not easily accessible ideas to establish. Present a variety of types: • sequences for which a rule is required; • rules from which a sequence can be generated.

For example, ask students to consider these sequences. • 1, 3, 5, 7, … • 2, 4, 6, 8, … • 1, 4, 9, 16, … • 5, 35, 245, … • 1, 3, 6, 10, … • 21, 31, 43, 57, … • 1, 1, 2, 3, 5, ... Ask: • For which of the sequences can you find rules of the following forms? un+1 = un + d un+1 = r un un+2 = a un+ 1 + b un un = f(n) In each case write down the values of the constants a, b, d, r, or the form of the function f(n).

More than one form of answer may be possible.

The different kinds of rule can be demonstrated on a spreadsheet. Students will need to understand the process of replication (in Excel, for example) and of how to express one cell as a function of another.

More able students could consider a formula for generating Fibonacci sequences. Derivation is beyond their scope but the appearance of √5 will intrigue them. The work could be linked to study of the golden section. For this, use the applet Golden rectangle at nlvm.usu.edu/en/nav/ vlibrary.html.

Unit 10A.8



Arithmetic sequences

Class discussion Present students with several arithmetic sequences and ask them for the common characteristics. Ask also if it is possible on the basis of what is given to work out, say, the hundredth term.

Draw out from the discussion: • the notion of first term; • common difference; • the goal of a formula for the nth term.

Establishing the nth term will inevitably focus on the fact that only (n – 1) differences have to be added to the first term to obtain it. Thus obtain the standard formula for the nth term.

The diagram is a useful representation of an arithmetic series.

Exercises From a list of sequences, students should identify the values of the first terms, and the common differences from the first two terms or from the first and a subsequent specified term. Once the notation is established, use written exercises to practise the use of the formula for the nth term.

More able students can establish the formula for the sum of n terms for themselves.

2.5 hours

Arithmetic sequences Identify and sum arithmetic sequences, including the first n consecutive positive integers, and give a ‘geometric proof’ for the formulae for these sums.

Sum to n terms

Class discussion Present the students with the problem of summing a large number of consecutive integers, such as 51 + 52 + … + 100. This problem can be tackled by a spreadsheet but takes time. The easy demonstration (see the notes on the right) prepares the way for the general result.

Exercises Exercises should include word problems that introduce arithmetic series in different contexts. Students need enough practice to avoid confusing the formula for the nth term with that for the sum of n terms.

51 + 52 + … + 100

100 + 99 + … + 51

151 + 151 + … + 151 = 50 × 151

So the sum is 25 × 151 = 3475.

Students should become familiar with the formulae:

un = a + (n – 1)d Sn = 1⁄2 [2a + (n – 1)d], where Sn stands for

the sum of the first n terms

5 hours

Geometric sequences Use mathematics to model and predict the outcomes of real-world applications.

Identify and sum geometric sequences and know the conditions under which an infinite geometric series can be summed.

Convert any recurring decimal to an exact fraction.

Geometric sequences

Class discussion

Begin by presenting three or four examples of geometric series. Ask students for the rule that defines the next term in each case. Establish the common features of this type of series. Draw out from students the formula for the general term based on the standard notation of a for the first term and r for the common ratio: un = arn–1.

Demonstrate how to find the sum for n terms, (1 )S(1 )

n

na r

r−=−

.

Sn = a + ar + ar2 + ar3 + … + arn–1

Multiplying by r gives: rSn = ar + ar2 + ar3 + … + arn–1 + arn Subtracting the second equation from the first: Sn(1 – r) = a – arn

(1 )S(1 )

n

na r

r−=−



Exercises These should begin with practice that: • poses different series so that students can distinguish those which are geometric from those

which are of other types (so that the concept of common ratio between all pairs of terms is made firm);

• rehearses the use of the formulae for the nth term and the sum of n terms in simple cases; • finishes with applications in word problems (where some APs can be mixed in).

Exercises on applications are important since they are opportunities for students to appreciate real-world applications. A good case is the idea of the chain letter as a means of making a fortune; the exploding general term for the geometric series shows the foolhardiness of the idea.

Geometric sequences that converge

Class discussion Point out that some geometric progressions converge while others do not. Explain the terms convergent, divergent, oscillatory. Give students the chance to distinguish them by the value of the common ratio. All should easily see that the criterion is whether or not | r | < 1. (This is a convenient moment to discuss modulus notation.) Then find the sum to infinity by deriving the limit from the sum to n terms:

S1ar

=−

The spreadsheet below can be used to help develop understanding. Columns A and B contain the terms and partial sums of a geometric series, defined by first term A1 and common ratio B1 (shaded); similarly for columns C, D and E, F. The lower pane shows the sequences and their sums further on. Assign values arbitrarily to B1, D1 and F1 to show the significance of the ratio’s value in affecting the convergence of the series.

Introduce more able students to the harmonic series: 1 + 1⁄2 + 1⁄3 + 1⁄4 + …

in which the terms decrease but the sum of the series itself diverges.

In the spreadsheet, the cell definitions are

A1 = 5

B1 = 4

A2 = A1 * B$1

B2 = A1 + A2

A3, etc, replicated from A2

B3 = B2 + A3

B4, etc, replicated from B3

C1 = 5

D1 = 0.4

E1 = 5

F1 = 1

A2:B101 copied to C2, E2



Exercises Confine practice initially to simple cases. After that, give practice in word problems and finally miscellaneous exercises that involve arithmetic and geometric series of all types considered.

Include applications, such as compound interest.

The simple cases should include recurring decimals considered as infinite convergent geometric series. Make links to students’ work on considering rational numbers as terminating or recurring decimals.

Problems • Grains of rice are placed on each square of a chessboard. The board has 64 squares.

One grain is placed on the first square, two on the second, four on the third, eight on the fourth, and so on. Calculate the total number of grains of rice on the chessboard. Given that 1 kilogram of rice contains approximately 16 000 grains of rice, estimate the weight of all the rice on the chessboard.

• In a race, skittles S1, S2, S3, …, are placed in a line, spaced 2 metres apart.

Sara runs from the starting point (b metres from the first skittle). She picks up the skittles, one

at a time and in order (S1, S2, S3, …), returning them to O each time (see figure). a. Show that the total distance Sara runs in a race with 3 skittles is 6(b + 2) metres. b. Show that the total distance she runs in a race with n skittles is 2n(b + n – 1) metres. c. With b = 5, the total distance she runs is 570 metres.

Find the number of skittles in this race. MEI

2 hours

Pascal’s triangle

Identify number patterns contained in Pascal’s triangle.

Preparing for the binomial theorem

Class discussion Revisit the identity (a + b)2 = a2 + 2ab + b2. Ask students what they expect of formulae such as (a + b)3 or (a + b)4. Students should suggest how the first result can be used to generate the second and third by successive multiplication by (a + b). Discuss how the multiplication can be carried out. Systematic tabular multiplication (as on the right) makes Pascal’s triangle easy to produce, as well as making the rule for generating one row from the previous one easy to derive. Lead students to this discovery.

2 2

3 2 2

2 2 3

3 2 2 3

2

22

3 3

a ab ba b

a a b aba b ab b

a a b ab b

+ ++

+ ++ +

+ + +



Investigation

As well as getting practice in applying Pascal’s triangle to binomial powers (so preparing for the binomial theorem), students can find a wealth of interest in the various patterns that can be noticed. The triangular numbers appear, for example, and the rows add to powers of 2.

Use the applet Pascal from the website nlvm.usu.edu/en/nav/vlibrary.html to discuss patterns in Pascal’s triangle.

Ask students to describe in words how any entry in Pascal’s triangle is related to entries in the row above. Set up an algebraic relationship to describe this. Ask: • Where are the triangular numbers located in Pascal’s triangle? • What other patterns can you spot?

Look at the numbers in an early row of Pascal’s triangle. Sum the squares of these numbers. Ask: • In what row is the answer located?

Identify where to find the sum of the squares of the numbers in any row of Pascal’s triangle. Get students to explain their reasoning.

More able students may wish to consider a formula for the general term of Pascal’s triangle.

The form !!( )!n

r n r− can be given for interest. Students can then verify for themselves the rule

for generating the next row from the previous one in the triangle.

There is a good website on Pascal’s triangle at mathworld.wolfram.com/PascalsTriangle.html.


Assessment


Each term of a sequence is 3 times the preceding term. The first term is 5. Set up a term-to-term definition for this sequence. Give an expression for the nth term in terms of n. Write down, but do not simplify, the 50th term.

The table shows the first six triangular numbers.

Position 1 2 3 4 5 6

Term 1 3 6 10 15 21

Investigate diagrammatic ways of representing these numbers.

Set up a relationship to describe the nth term in terms of its position value n. What is the 100th triangular number? What is the 1000th triangular number?

Find the sum of the first n consecutive positive integers, and hence the sum of any set of n consecutive positive integers.

Find the sum of all numbers between 1 and 100 that are exactly divisible by 3.


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

Explain why 40.12 .33• •

=

A geometric progression is defined by uk = 3 × 1.25–k, k = 1, 2, 3, …

a. Calculate u1, u2 and u3. What is the common ratio of the geometric progression?

b. Calculate 20

1u .k

k =∑

c. Find the sum to infinity of the geometric progression.

MEI

For each of the following sequences, state whether it is arithmetic, geometric or neither of these. For those that are arithmetic or geometric, find the sum of the first 20 terms of the corresponding series.

a. 50, 52, 54, 56, … b. un = 2 × 0.8–n c. un = 2n + 3 d. un = n2 e. un+1 = –un, u1 = 2 f. un+1 = 2un + 1, u1 = 1

In the case of one of these sequences, the corresponding series has a sum to infinity. Calculate the sum to infinity of this series.

MEI

Unit 10A.8



Quadratic functions and their graphs

About this unit This is the third of three units on algebra for Grade 10 advanced. It builds on Units 10A.3 and 10A.6, Algebra 1 and 2.





Previous learning To meet the expectations of this unit, students should already be able to solve quadratic equations of the form a2x2 – b2y2 = 0 or x2 ± 2ax + a2 = 0 by factorisation. They should be able to plot graphs of simple quadratic and cubic functions and use graphical methods to find approximate solutions of quadratic equations, on paper and using ICT.

Expectations By the end of the unit, students will be aware of the role of symbols in algebra. They will generate and manipulate algebraic expressions, including algebraic formulae. They will use algebraic methods to solve quadratic equations. They will plot and interpret quadratic graphs. They will find the tangent at a point on the graph of a function. Through their study of quadratic functions and their graphs, and the solution of the related equations, they will begin to appreciate numerical and algebraic applications in the real world. They will use ICT to analyse problems.

Students who progress further will acquire increasing facility with algebraic notation. They will use the determinant to consider the nature of the roots of a quadratic equation and, where possible, will find the approximate solution by graphical methods. They will find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation. They will understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c. They will use quadratic functions to model a range of situations.



• spreadsheet software such as Microsoft Excel • computers with Internet access, graph plotting and spreadsheet software

for students • graphics calculators

Key vocabulary and technical terms Students should understand, use and spell correctly: • quadratic, monomial, binomial, polynomial, factorisation, root, completing

the square • parabola, tangent, gradient, maximum, minimum, stationary value

UNIT 10A.10 14 hours






10A.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

10A.1.2 Use mathematics to model and predict the outcomes of real-world applications.

10A.1.3 Identify and use interconnections between mathematical topics.

10A.1.5 Use a range of strategies to solve problems.

10A.1.9 Generalise whenever possible.

10A.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.

10A.1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.

11A.5.5 Find approximate solutions of the quadratic equation ax2 + bx + c = 0 by reading from the graph of y = ax2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis.

9.3.8 Solve quadratic equations of the form a2x2 – b2y2 = 0 or x2 ± 2ax + a2 = 0 by factorisation; verify the solutions by substituting in the original equation.

10A.4.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

10A.5.15 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 or decreasing at the point, or stationary.

9.4.4 Generate points and plot graphs of simple quadratic and cubic functions, e.g. y = 3x2 + 4, y = x2 – 2x + 1; use graphical methods to find approximate solutions of quadratic equations, on paper and using ICT.

10A.5.16 Translate the statement y is proportional to x2 into the symbolism y ∝ x2 and into the equation y = kx2 and know that the graph of this equation is a parabola through the origin.

11A.5.4 Given a quadratic equation of the form ax2 + bx + c = 0, know that: • the discriminant Δ = b2 − 4ac must be non-

negative for the exact solution set in to exist; • there are two distinct roots if Δ is positive and

one repeated root if Δ is zero.

10A.5.19 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 identify the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when quadratic functions are increasing, when they are decreasing and when they are stationary.

11A.5.3 Find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

2 hours

Solution of quadratic equations by factors

2 hours

Quadratic equations – problems

2 hours

Solution of quadratic equations by completing the square

3 hours

Solution of quadratic equations by formula

5 hours

Quadratic graphs

10A.5.20 Model a range of situations with quadratic functions of the form y = ax2 + c.

11A.5.2 Model a range of situations with appropriate quadratic functions.

Unit 10A.10


Activities


Using factors to solve quadratics

Class discussion Students met quadratic equations with two real roots in Grade 9 and will have found approximate solutions by using graphs. This unit builds on that work. Students are introduced to an analytic method of solving quadratic equations, which, at least in the first instance, leads to exact answers. This motivates further practice on factors, so building on the work of Unit 10A.6, Algebra 2.

Introduce the basic idea with some questions. Ask students: • What can be deduced if 4x = 0? • What can be deduced if ab = 0? • What can be deduced if (x – 2)(x – 3) = 0? • What can be deduced if x2 – 5x + 4 = 0? • What can be deduced if x2 – 3x = 10?

Draw out a general procedure of: • recognising a quadratic equation by the presence of x2 or equivalent; • rearranging the equation so that the right-hand side contains just zero; • factorising the left-hand side; • considering each factor in turn when set equal to zero; • checking by substituting into the original equation.

Discuss particular cases that tend to prove troublesome: • s2 – 4s = 0 (forgetting about the common factor); • s2 – 4 = 0 (guessing that s = 2 only and forgetting the negative root).


2 hours

Solution of quadratic equations by factors

Solve quadratic equations exactly, by factorisation.

Exercises Give students plenty of practice: • to reinforce factorisation skills; • to keep in sight techniques for solving linear equations.

Remind students that factors still need to be checked carefully.

After some elementary practice, review more complex simplifications from Unit 10A.3. Consider, for example:

2 2

( 2)( 4) ( 1)( 2) 06 8 3 2 0

9 6 0

x x x xx x x x

x

+ + − − − =⇒ + + − + − =⇒ + =

which is not a quadratic at all, despite appearances.

Avoid equations that have no real roots at this stage. The purpose is that students appreciate that this method produces exact answers (in contrast to the use of the formula in later practice).

Unit 10A.10



2 hours

Quadratic equations – problems Solve routine and non-routine problems in a range of mathematical and other contexts, including closed problems.

Use mathematics to model and predict the outcomes of real-world applications.

Problem solving with quadratics

Class discussion Quadratic equations open up new possibilities for problems. Return to the idea of modelling, which is not a method but more a style of approach. In most problems a diagram will help. In every problem the first step is to specify the unknown quantity to be used (where this is not stated in the question) in a precise way, including any units associated with it.

Demonstrate several different problems on the board.

Class exercises Give practice in a range of problems. Less confident students may require support to get under way, but insist that all students make some sort of start on any particular problem before you offer assistance – by specifying the unknown or by drawing an appropriate diagram, for example.

Problems • In three years time Roza’s age will be the square of her age of three years ago.

How old is she? • Is the mean of the squares of two numbers greater than, or less than, the square of their

means? • Is it always true that if you double the sum of two squares you get the sum of two squares?

If so, can you prove it? Here are some examples: 2(52 + 32) = 2(25 + 9) = 68 = 64 + 4 = 82 + 22

2(72 + 42) = 2(49 + 16) = 130 = 121 + 9 = 112 + 32

Example problem A rectangular lawn is 22 metres by 16 metres. It is surrounded by a path w metres wide. The path requires 1472 kg of coarse sand to surface; the supplier recommends 4 kg per square metre. If the supplier’s instructions have been rigorously followed, how wide is the path?

Solution

Measurements in metres

2

2

2

(22 2 )(16 2 ) 22 16 1472 4368

352 76 4 352 368 04 76 368 0

19 92 0( 4)( 23) 0

4 or 23

w w

w ww w

w ww ww w

+ + − × = ÷=

⇒ + + − − =⇒ + − =⇒ + − =⇒ − + =⇒ = = −

From the context, w > 0, so we conclude that the width of the path is 4 metres.

2 hours

Solution of quadratic equations by completing the square

Solve quadratic equations exactly by completing the square.

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

Completing the square

Class discussion Introduce completing the square.

Review the expansion for (a + b)2. Stress its structure as ‘square of the first’, ‘twice the product’ and ‘square of the second’. Practise this with a few examples so that it is fully reinforced.

Then pose the questions: • Is x2 + 22x + 121 a square? • Is x2 + 30x + 525 a square? • How can you tell? • What must be added to x2 + 14x to make it a completed square?

Students will then be ready to appreciate the completion of the square argument.

Demonstrate two or three examples, as shown on the right.

Example 2

2

2

2

6 5 06 56 9 5 9

( 3) 14

3 14

3 14

x xx xx x

x

x

x

− − =⇒ − =⇒ − + = +⇒ − =

⇒ − = ±

⇒ = ±

This presentation parallels exactly the derivation of the quadratic formula to come next.



Exercises Get students to practise the technique of completing the square. They will need calculators to finish the arithmetic. Solutions should be finished tidily with roots to a prescribed degree of accuracy.

Avoid setting questions which have exact roots since this can undermine the work done on solution by factorisation (even if one example may be done to show that the same answers are obtained).

The real value of completing the square is its algebraic significance in later work.

3 hours

Solution of quadratic equations by formula

Solve quadratic equations by using the quadratic formula.

Solution by formula

Class discussion Students are now ready for the quadratic formula.

The formula is not only useful but closely connected to completing the square, and, even if some students find the derivation hard to follow, they should appreciate enough of the argument to remember how it was derived.

The derivation is best done as a parallel of a worked example using the method of completing the square. The derivation on the right is done with exactly the same steps and arrangement as the example for completing the square.

Show how to use the formula by applying it to one or two examples.

2

2

2

2 22

2 2

2

2

2

2

2

2

0 ( 0)

0

(since 0)

2 2

2 44

44

2 4

42

ax bx c ab cx xa ab cx x aa a

b b c bx xa a a a

b c bxa a a

b aca

b b acxa a

b b acxa

+ + = ≠

⇒ + + =

⇒ + = − ≠

⎛ ⎞ ⎛ ⎞⇒ + + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−⎛ ⎞⇒ + = +⎜ ⎟⎝ ⎠

−=

−⇒ + = ±

− ± −⇒ =

Exercises After deriving the formula, give students some practice. They will need calculators. To avoid mistakes, encourage them to write down: • the equation to be solved; • the assignation of a, b and c; • the formula in its standard form; • the substituted numbers.

Encourage students to complete the calculation using their calculators and to write down the answer rather than work through intermediate stages of calculation. This avoids the bad practice of premature rounding, especially of the approximation for the square root.

For the evaluation of the square-root argument, a common error arises from an incorrect sign for the 4ac term. Suggest that the best way to avoid this is to work successively through ac, 4ac, –4ac.

Make the point that factorisation is usually shorter and less error-prone where it can be done. In examination questions, it is usual to find hints that the formula is to be used since the question will often refer to the need for approximation (by requiring answers to two decimal places, for example).

More able students may be interested in Cardan’s formula for cubics. The formula is quoted on www.math.vanderbilt.edu/~schectex/ courses/cubic.



Miscellaneous practice To see the topic as whole, set mixed examples of exact and approximate types, as well as questions that are set in context or in words.

5 hours

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


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

Use a range of strategies to solve problems.

Identify and use interconnections between mathematical topics.

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 or decreasing at the point, or stationary.

[continued]

Working with graphs of quadratic functions

Investigation Students have met quadratic graphs in Grade 9, so they should not need to rehearse techniques for plotting smooth curves. In any case, smooth curves occur later in this unit (in work on tangents).

In unit 10A.6, students investigated straight line graphs by using a graphics calculator or Autograph. They should be ready to investigate parabolas and their relationship to functions of the form y = ax2 + bx + c.

Set this up as follows. • Use your graphics calculator (or Autograph) to investigate y = ax2 + bx + c. • Obtain a graphics display of the curves you get by setting a = 1, b = 2, and varying c. • Write a sentence to summarise the significance of c. • With your own (different) choice of a and b, see if the same result follows in the different

case. • Obtain a graphics display of the curves you get by setting b = 2, c = 1 and varying a. • Write a sentence to summarise the significance of a. • Check that your conclusion is still valid if you choose your own values of b and c. • Obtain a graphics display of the curves you get by setting a = 1, c = –1 and varying b. • Try to determine values of a, b and c so that the resulting curve passes through each of the

lowest points of the curves in the last display.

Try to draw out general conclusions about a and c from students’ contributions. Establish that: • the value of a affects the width-scale of the curve (and negative values invert it); • the value of c has the same effect as in the case of a straight line.

Avoid describing examples of the same value of a as ‘parallel’.

At this point you could consider the special cases of b = 0 and c = 0. Translate the statement y is proportional to x2 into y ∝ x2 and the equation y = ax2. It should be evident that the graph of this equation is a parabola through the origin.

A second investigation can focus on the significance of b. This can be done by tabulating the equation of the axis of symmetry (by inspection of a graphics display) against the original equation. Students should quickly conclude that the axis is always at x = –1⁄2 b. More able students should be able to generalise this for the case that a is not 1. Highlight the link with the quadratic formula.



Try a third approach to the topic in which students are invited to assign equations (i.e. determine a, b and c) to match parabolas. This can be done in Excel using randomly generated integers to generate curves, or manually in Autograph.

For an example, see the one on the right.

Example On the graphs shown below, a = ±1 in every case. What are the values of b and c for each curve?

[continued] Translate the statement y is proportional to x2 into the symbolism y ∝ x2 and into the equation y = kx2 and know that the graph of this equation is a parabola through the origin.

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 identify the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when quadratic functions are increasing, when they are decreasing and when they are stationary.

Model a range of situations with quadratic functions of the form y = ax2 + c.

Class discussion From their work with graphics calculators and Autograph, students should be familiar with the sort of picture to expect from quadratic functions. They will also be familiar with the gradient of straight lines.

Discuss the gradient of curves.

Show students a distance–time graph for rockets launched into the air, such as the one on the right. Explain to the class that when they are faced with a graph, they should ask themselves simple questions such as: • What part of the graph corresponds to the moment at which the rocket is launched? • What is the time of flight? • What is the maximum height?

Take different sections of the curve, and consider the heights and times which correspond. Draw out that different parts of the curve correspond to different speeds. This leads to further questions: • Which part of the graph corresponds to the fastest speed? • Which corresponds to the slowest?

Make sure that students appreciate that the steepness (and hence gradient) measures speed. Ask students to suggest how to measure speed at a particular instant. Draw out that this is so by calculating the gradient of a tangent drawn to the curve.

Autograph can be used to make the point that gradient is itself a function. The lower graph could serve as the distance–time graph just considered. The sloping line corresponds to the value of the gradient at each point.

Introduce students to the terminology of gradients and associated features of graphs, such as maximum point, stationary point, turning point.

Reinforce these in the exercises that follow.



Exercises With the availability of calculators and computer applications, keep the plotting of graphs and the drawing of tangents to the minimum necessary to grasp the ideas. A gradient measured in this practical way is inevitably prone to error, deriving from the standard of plotting and drawing of the original graph. It is better to make full use of the gradient capabilities of the software.

Examples Use Autograph (or your graphics calculator) to show the graph y = x2 – 4x + 3. From your display: • state the equation of the axis of symmetry; • estimate the coordinates of the minimum turning point; • estimate the roots of the equation 0 = x2 – 4x + 3; • solve approximately the equation 0 ≥ x2 – 4x + 3; • estimate the gradient of the curve where x = 1.

Use Autograph (or your graphics calculator) to show the graphs y = 2x2 – 4x + 3 and y = 1 – 4x – 3x2. From your graph: • estimate the roots of the equation 0 = 1 – 4x – 3x2; • estimate the coordinates of the points where the two curves cross; • find an equation which must be solved to determine exactly the x-coordinates of the points

where the two curves cross; • solve the equation, giving your roots to two decimal places; • estimate the gradient of each curve at the two points where they cross (so you will have four

answers).


Assessment


a. Factorise the expression x2 – 13x + 36.

b. Using your answer to (a), solve x2 – 13x + 36 = 0.

c. Solve x2 – 13x + 36 > 0.

MEI

a. Factorise 15x2 + 13x – 6.

b. Solve the equation 15x2 + 13x – 6 = 0.

c. The equation x2 + mx + n = 0 has the solution x = 2 or x = –1. Find the values of m and n.

MEI


Find values of p and q (in each case) such that:

• x2 + 3x = (x + p)2 + q

• x2 – 7x = (x + p)2 + q

Deduce that x2 + 3x ≥ – 9⁄4 and find the least value of x2 – 7x.

Solve each of these inequations, marking the solution sets on a number line.

• x2 – 4x > 0

• x2 – 4 > 0

• 5 – 2x – 3x2 ≥ 0

The sketch shows the graph of the function x2 + hx + k. Determine the values of h and k.

Create a display like this with your graphics calculator.

Unit 10A.10



The graph shows four curves, each of which is a quadratic function. One of them is y = –x2 + 4x – 2. Identify the other three.

If I think of an integer, square it and add six, the result is the same as doubling it, adding five, squaring the result, and subtracting seven. What is the number I am thinking of?

Use squared paper and a scale of 10 cm per unit on the h-axis and 20 cm per unit on the t-axis to plot the graph of h = 5t – 4.9t2 for values of t from 0 to 1.1. This graph can be used to model the height, h metres, of a stone thrown vertically upwards at 5 m s–1, where t seconds is the time from the moment of throwing. From the graph:

• estimate the time at which the stone returns to the point from which it was thrown;

• estimate the maximum height and the time at which it reaches it;

• estimate the time for which the stone is above 1 metre from its throwing point;

• estimate the velocity after 0.5 seconds.


GRADE 10A: Geometry and measures 2

Developing Pythagoras’ theorem

About this unit This is the second of two units on geometry and measures for Grade 10 advanced. It builds on Unit 10A.5, Geometry and measures 1.





Previous learning To meet the expectations of this unit, students should already know and be able to use the sine, cosine and tangent ratios to solve right-angled triangle problems with the aid of a scientific calculator. They should be able to state and apply Pythagoras’ theorem, but without proof.

Expectations By the end of the unit, students will identify and use interconnections between mathematical topics. They will use their knowledge of geometry, Pythagoras’ theorem and the trigonometry of right-angled triangles to solve practical and theoretical problems relating to shape and space. They will be familiar with the Cartesian equation of a circle.

Students who progress further will solve complex trigonometric problems by breaking them into smaller steps. They will use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°. They will know and use the sine rule and the cosine rule to solve triangle problems in two and three dimensions.

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



• calculators for students • tape measure, theodolite

Key vocabulary and technical terms Students should understand, use and spell correctly: • words related to triangles: hypotenuse, adjacent, opposite, Pythagoras • words related to trigonometry: tangent, sine, cosine • words related to functions and graphs: Cartesian, inverse (as applied to

circular functions) • words related to problem solving and reasoning: proof, converse

UNIT 10A.11 10 hours






10A.1.3 Identify and use interconnections between mathematical topics. 11A.1.4 Break down complex problems into smaller tasks.

9.7.3 Know the sine, cosine and tangent ratios for a right-angled triangle.

11A.8.2 Know and use the sine rule and the cosine rule to solve triangles.

9.7.2 Solve problems involving finding a side of a right-angled triangle.

11A.8.3 Solve triangle problems in two and three dimensions.

9.7.4 Use a scientific calculator to: • find the values of trigonometric

ratios; • find an angle using the inverse

trigonometric function keys.

10A.6.5 Know the standard trigonometric ratios, and their standard abbreviations, for sine of θ, cosine of θ and tangent of θ, given an angle θ in a right-angled triangle, and use these ratios to find the remaining sides of a right-angled triangle given one side and one angle or to find the angles given two sides.

9.7.1 State and apply Pythagoras’ theorem (not proof).

10A.6.6 Derive and recall the exact values for the sine, cosine and tangent of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations.

10A.6.7 Discuss at least two proofs of Pythagoras’ theorem.

4 hours

Pythagoras’ theorem

4 hours

Elementary trigonometry (with Pythagoras)

2 hours

The equation of a circle (including the distance formula)

10A.6.8 Use Pythagoras’ theorem to find the distance between two points, to

solve triangles, to find Pythagorean triples, and to set up the Cartesian equation of a circle of radius r, centred at the point (α, β).

11A8.7 Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°.

Unit 10A.11


Activities


Proving Pythagoras’ theorem

Class discussion Students are accustomed to using Pythagoras’ theorem, and to applying it in calculations. This unit introduces a rigorous proof. Although the foundations of geometry have been stressed in earlier geometry units in Grade 10, there will still be a certain haziness associated with facts which have been established (like the equality of base angles of an isosceles triangle), and facts which are taken on trust (like V = 1⁄3 πr2 h). Remind students of the need for proof, distinguishing between assumptions, definitions and theorems.

There are many ways to prove Pythagoras’ theorem. The choice is determined by two factors: • establishing the result as a consequence of argument and students’ current knowledge; • demonstrating different styles of proof to show the connections between different parts of

mathematics.

Here are two possible approaches. Others are referenced in the website in the notes.

Other proofs (and other material relating to Pythagoras’ theorem) can be viewed at Pythagorean theorem (http://www.cut-the-knot.org/pythagoras/index.shtml).


4 hours

Pythagoras’ theorem Identify and use interconnections between mathematical topics.

Discuss at least two proofs of Pythagoras’ theorem.

Derive and recall the exact values for the sine, cosine and tangent of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations.

Use Pythagoras’ theorem to find Pythagorean triples. Algebraic proof

The right-angled triangle ABC (1) is replicated by three copies (2, 3 and 4). The resulting figure is a square because it has a right angle at each corner. Each side is the same length, because each combination x + y has the same length. Since CEGA is a quadrilateral with four equal angles, r = 90°, and since p + q = 90°, p + q + r = 180°, making BCD a straight line. ACEG is also a square. Then by consideration of areas: 4 × ABC + square ACEG = square BDFH

Hence:

2 212

2 2 2

2 2 2

4 ( )

2 2

xy z x y

xy z x xy yz x y

× + = +

⇒ + = + +⇒ = +

There are other proofs based on the same jigsaw puzzle idea.

Use the applet Proof without words: Pythagorean theorem (proof 2) to illustrate a similar but slightly different approach (see illuminations.nctm.org/tools/index.aspx). In this case, the area of the large square (c2) is shown to be equivalent to the area of two rectangles (2ab) plus the area of the small square (a2).

Unit 10A.11


Objectives Possible teaching activities Notes School resources A geometric proof

In the figure on the right, the right-angled triangle ABC has squares AA′B′B drawn on AB, BB′′C′′C on BC, and CC′′′A′′′A on CA. A perpendicular has been dropped from B to E, cutting AC at D.

Now for the proof.

Triangle ACA′ is a rotation of triangle AA′′′B through 90°, so the two will be congruent; this can be shown by the test SAS, using sides of squares and equal corresponding angles at A. By the same argument triangle AC′′C is congruent to triangle C′′′BC. Now triangle ACA′ has half the area of square AA′B′B, since its base and height are each sides of the same square; similarly triangle ABA′′′ has half the area of rectangle ADEA′′′.

Thus square AA′B′B has the same area as rectangle ADEA′′′.

Similarly, square BB′′C′′C has the same area as rectangle DCC′′′E. Thus by areas: square AA′B′B + square BB′′C′′C = rectangle ADEA′′′ + rectangle DCC′′′E = square AA′′′C′′′C

This establishes Pythagoras’ theorem.

Investigations • Make a replica of the five parts of the diagram used for the algebraic proof, i.e. four congruent

right-angled triangles and the internal square. Find another way of rearranging them to prove Pythagoras’ theorem. (For an example, see the applet Proof without words: Pythagorean theorem (proof 2) at illuminations.nctm.org/tools/index.aspx)

• Get students to tackle the problems in the interactive applet Pythagorean theorem at nlvm.usu.edu/en/nav/vlibrary.html.

• Sketch a right-angled triangle ABC in which AB = AC. Which angle is a right angle? Find the values of the two other angles. If AB = AC = 1, calculate BC. What exact values for sine, cosine and tangent does this enable you to obtain?

• Sketch an equilateral triangle ABC. Draw in its altitude AH. What can you say about BH and CH? What can you say about ∠ABC? If AB = BC = CA = 2, find exact values for sin 60°, cos 60° and tan 60°. What other exact sine, cosine and tangent can you find from the same figure?

• Show that (m2 + n2)2 – (m2 – n2)2 = 4m2n2. Why does this mean that the three numbers m2 + n2, m2 – n2, 2mn, where m > n > 0, can be measurements of the sides of a right-angled triangle? Investigate the triples you get for different integral values of m and n.

Discuss students’ approaches and solutions to these problems.

Note: The investigation into Pythagorean triples implicitly uses the converse of Pythagoras’ theorem, but not all students will have realised it – does a2 + b2 = c2 for the sides of a triangle imply there is a right angle opposite c? This is taken up in the next section.

Extension for more able students A hard challenge for those who work rapidly through these investigations is as follows.

• x = 4, y = 4, z = 2 is a solution of 1 1 1x y z

+ = .

Can you find another, where x, y and z are still positive integers?

Can you find other solutions where x ≠ y (and are still positive integers)?

Can you find all positive integer solutions?



Class discussion The last investigation touched on the converse of Pythagoras’ theorem. Ask students to show that 252 = 242 + 72. Then ask: • If a triangle has sides 7, 24 and 25, must it be right-angled?

Some but not all students may realise that the converse of Pythagoras’ theorem cannot just be assumed and that it has to be proved.

Here is a proof.

If a, b, c satisfy a2 + b2 = c2, then draw a triangle with these three lengths. Draw another triangle with sides a and b and an included right angle. By Pythagoras, the third side (opposite the right angle) must be c. The two triangles are now congruent by SSS, so the first has a right angle too.

Linking Pythagoras’ theorem and trigonometry

Finding sides: practical work If students have not done so before, begin the use of trigonometry by applying it to a practical situation. Focus on finding the height of a building whose top or stairway is inaccessible (so that dangling a stone on the end of a string will not work). This requires a long tape measure or surveyor’s tape and a theodolite (or means of measuring angle of elevation – one can easily be made with cardboard). This does not take long to organise and makes the point that trigonometry has many practical applications.

Class discussion Revise sine, cosine and tangent and three important aspects of elementary trigonometry: • the techniques apply (in the early stages) to right-angled triangles only; • there is a convention for the naming of the sides of a right-angled triangle – the hypotenuse,

the opposite and the adjacent – with respect to a given angle; • every calculation should be based on a good diagram.

Show students that sintancos

θθθ

= .

Discuss the solution of right-angled triangles where the problem is to find a single side, given another side. Here is an example of the type of problem and the approach.

4 hours

Elementary trigonometry (with Pythagoras) Identify and use interconnections between mathematical topics.

Know the standard trigonometric ratios, and their standard abbreviations, for sine of θ, cosine of θ and tangent of θ, given an angle θ in a right-angled triangle, and use these ratios to find the remaining sides of a right-angled triangle given one side and one angle or to find the angles given two sides.

Use Pythagoras’ theorem to solve triangles.

Encourage students to: • draw the diagram; • name the sides; • decide which ratio to use.

Here is a model calculation.

tan3777 tan375.2748...5.27, to 3 s.f.

x

x

° =

⇒ = °==

Useful demonstrations of trigonometry at work and tutorials for students are on MathsNet (www.mathsnet.net/asa2/2004/backtrig.html).



Exercises Include examples which put these techniques into practice, covering the following: • simple cases, just like the examples, but mixing sin, cos and tan; • cases in which the problem is just as simple but the triangle’s orientation is non-standard; • cases in which the unknown side first appears in a denominator; • use of the converse of Pythagoras’ theorem to check for a right angle.

Use this as an opportunity to check that all students can competently manage their calculators.

Finding angles

Class discussion Use the same approach to find an angle, a technique that students should have met in Grade 9. Describe arc to students as equivalent to the shift or function key on a calculator when used in conjunction with sin, cos and tan keys.

Here is a further model calculation.

4tan7

4arc tan7

29.74...29.7 to 1 d.p.

x

x

° =

⇒ ° =

== °

More able students may be introduced to the trigonometric equivalent of Pythagoras’ theorem: cos2 θ + sin2 θ = 1.

Exercises The new technique should be practised and used alongside what has gone before. The work should include: • simple cases, just like the example, but mixing sin, cos and tan; • cases in which the problem is just as simple but the triangle’s orientation is non-standard; • cases in which Pythagoras’ theorem is either advantageous or necessary; • problems which implicate isosceles triangles and where it is necessary to introduce a right-

angle by construction; • miscellaneous problem solving with more complex diagrams and multi-step solutions.

Example Solve the triangles shown, giving all the angles and all the sides.



2 hours

The equation of a circle (including the distance formula)

Use Pythagoras’ theorem to find the distance between two points, and to set up the Cartesian equation of a circle of radius r, centred at the point (α, β).

Identify and use interconnections between mathematical topics.

Equations of circles

Class discussion Pose the problem of finding the distance between two points in the coordinate plane – not both on the same abscissa or same ordinate. Show students that the key is a diagram which makes the x- and y-steps the sides of a right-angled triangle, followed by use of Pythagoras’ theorem.

Once that approach has been grasped, it can be formalised in the distance formula for the distance between (x1, y1) and (x2, y2):

2 21 2 1 2( ) ( )x x y y− + −

Introduce the Cartesian equation of a circle as an application of the distance formula. Develop the equation of a circle from first principles each time, using the distance formula to equate the distance between the specified centre and a general point (x, y).

The reverse problem of recognising the equation of a circle and manipulating it to find its centre and radius is also possible by formulae, but again an approach from first principles is better, as an example of completing the square. Use the opportunity to link this work with algebra.

Introduce students who show interest to Cartesian as a name for a coordinate system that is an alternative to the polar coordinate system.

The equation of a circle of radius 7 and centre (4, 5) can by found by using the distance formula and algebraic techniques:

2 2 2

2 2

2 2

( 4) ( 5) 78 16 10 25 49

8 10 8

x yx x y y

x y x y

− + − =⇒ − + + − + =⇒ + − − =

The reverse problem – finding the centre and radius for a given circle’s equation – can be tackled by completing the square.

( )( )

2 2

2 2

22 2 2152

2 2152

2 215 6252 4

15 10 7515 10 8

15 10 5

75 5

( ) ( 5)

x y x yx x y y

x x y y

x y

+ − + =⇒ − + + =

⇒ − + + + +

= + +

⇒ − + + =

So this is a circle of centre (71⁄2, –5) and radius 121⁄2.

Exercises These should develop as follows: • practice on the distance formula; • practice on finding the equation of a circle; • finding the centre and radius given the equation of a circle; • calculations on the tangent kite, which combine Pythagoras’ theorem, the distance formula

and the equation of a circle; • some problem solving with more complicated configurations requiring initiative.


Assessment


Find the equation of a circle of radius 5 units, centred at the point (5, –3).

Find the exact distance between the point (1, 4) and the point (–2, 5).

Each side of a cube is 5 cm. Calculate the length of a diagonal of the cube from one vertex on the ‘base’ to the opposite vertex on the ‘top face’. What is the angle between this diagonal and the base?

In this question, you should not use a calculator.

An elastic band is fixed on four pins on a pinboard, as shown in the diagram. Show that the total length of the band in this position is 14√ 2 units.


Show that a triangle with sides of length m2 – n2, 2mn and m2 + n2 respectively is always right-angled. Find some right-angled triangles using this result.

Suggest possible equations for these straight lines. Find the shortest distance between them.

ABCD is part of the design of a roof. ADC is horizontal and BD is vertical. The dimensions are as shown in the diagram in metres.

a. Calculate the size of the angle ABD.

b. Calculate the length of CD.

MEI (part question)

A ladder is placed against a vertical wall. The ground is horizontal and the foot of the ladder is 120 cm from the ground. The foot of the ladder is moved 30 cm further away from the wall.

Calculate how far the top of the ladder moves down the wall.

MEI

Not to scale

Unit 10A.11

Documents

Mathematics units Grade 10 advanced - csomathscience · PDF fileMathematics units Grade 10 advanced Contents 10A.1 Number 33 10A.7 Geometry 2 87 10A.2 Geometry 1 41 10A.8 Number and