Mathematics units Grade 11 foundation

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1 Mathematics units Grade 11 foundation Contents 11F.1 Number F.8 Number and algebra F.2 Geometry F.9 Geometry and measures F.3 Algebra F.10 Algebra F.4 Geometry and measures F.11 Statistics F.5 Statistics F.12 Geometry F.6 Algebra F.13 Algebra F.7 Geometry and measures F.14 Geometry and measures 4 203

2 Mathematics scheme of work: Grade 11 foundation units 90 teaching hours 1st semester 45 hours UNIT 11F.0: Grade 10F revision 3 hours UNIT 11F.5: Statistics 1 Designing a statistical enquiry Sampling and collecting data Representing data: stem-and-leaf diagrams; relative frequency histograms; cumulative frequency distributions; box plots 6 hours 2nd semester 45 hours UNIT 11F.11: Statistics 2 Representing and interpreting data Measures of central tendency Using ICT to analyse large data sets Making inferences and presenting findings using graphs, charts and tables 7 hours UNIT 11F.1: Number Fractions, decimals and percentages Proportional reasoning Laws of indices 7 hours UNIT 11F.8: Number and algebra Algebraic manipulation Expressions and formulae Geometric sequences and series 8 hours UNIT 11F.3: Algebra 1 Proportionality Properties of quadratic functions and their graphs 7 hours UNIT 11F.6: Algebra 2 Properties of graphs of functions Properties of linear functions Inequalities 9 hours UNIT 11F.10: Algebra 3 Properties of quadratic functions and their graphs Solution of quadratic equations by algebraic and graphical methods 9 hours UNIT 11F.13: Algebra 4 Inverse proportion Solution of simultaneous equations (one linear and one quadratic) by algebraic and graphical methods 8 hours UNIT 11F.4: Geometry and measures 1 Trigonometry; Pythagoras' theorem Solving triangles Sine and cosine rules 6 hours UNIT 11F.7: Geometry and measures 2 Applications of Pythagoras' theorem Cartesian equation of a circle 4 hours UNIT 11F.9: Geometry and measures 3 Circular functions and graphs Trigonometry 5 hours UNIT 11F.14: Geometry and measures 4 Compound measures; rates; SI units Circle problems; radians; bearings; latitude; longitude; great circles 4 hours UNIT 11F.2: Geometry 1 Using ICT to explore geometric proof 3 hours UNIT 11F.12: Geometry 2 Circle theorems Proof 4 hours Reasoning and problem solving should be integrated into each unit 15% 55% 30%

3 GRADE 11F: Number Calculations UNIT 11F.1 7 hours About this unit This is the first of six units on number and algebra for Grade 11 foundation. It builds on the work on number in Grade 10 foundation. 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 10 foundation or Grade 12 foundation. 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 understand exponents and roots, and be able to use the laws of indices to simplify expressions. They should be able to sum arithmetic sequences. They should be able to use percentages, ratios, scale and enlargements to solve problems. Expectations By the end of the unit, students will use the laws of exponents, proportional reasoning and harder percentage calculations to solve problems, including compound interest problems. They will convert any recurring decimal to a fraction. They will solve routine and non-routine problems in mathematical and other contexts using a range of strategies. Students who progress further will appreciate a wide range of numerical applications in the real world and will develop further confidence with the skills practised in the unit. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector computers with Internet access for students calculators for students Key vocabulary and technical terms Students should understand, use and spell correctly: exponent, index, indices, surd, standard form proportion, conversion percentage, compound interest, recurring decimal, fraction, denominator, numerator 115 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

4 Standards for the unit Unit 11F.1 7 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 11F and 12F standards 3 hours Rules of indices 10F.3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the x y key on a calculator. 11F.3.1 Understand and use the laws of exponents to calculate and simplify problems, including mental calculations in appropriate cases. 12F.3.1 Develop further confidence in all the calculation skills established in Grades 10 and hours Proportional reasoning 2 hours Fractions, decimals and percentages 10F F.3.6 Sum arithmetic sequences, including the first n consecutive integers, and give a geometric proof for the formulae for these sums. 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. 11F.3.2 Solve a range of problems using the multiplicative nature of proportional reasoning. 10F.3.7 Perform percentage calculations, including finding a percentage of a percentage and inverse percentages. 11F.3.3 Perform harder percentage calculations, including taking a percentage of a percentage, inverse percentage and compound interest problems. 11F.3.4 Investigate the problem of compounding interest more and more frequently and note that this tends to a limiting value; use this context to learn about the number e. 11F.4.2 Convert any recurring decimal to an exact fraction. 11F.1.5 Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary. 116 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

5 Activities Unit 11F.1 Objectives Possible teaching activities Notes School resources 3 hours Rules of indices Understand and use the laws of exponents to calculate and simplify problems, including mental calculations in appropriate cases. Problems with exponents Set students problems that allow them to build confidence and fluency with handling calculations with exponents (including surds, indices and standard form) in a range of realworld and abstract contexts. Encourage them to discuss their reasoning and to share their ideas with one another. For example: Write the following in order of magnitude: 8, 4 3, 5 3. Explain why ( 4) is not a real number. Which is larger, or ? Why? What value should be given to 0 0? Why? Evaluate 16 3/4. Discuss why ( ) = 2 8 = 2 4 = Find the value of k in x k = x 2 (x 3 ). Find the value of k in 2 k = Find the value of x for which 27 x = 9 5 x. Find the lengths of the sides of the squares in the diagram on the right and write them in their simplest exact form. Practice Supplement with extra demonstrations and practice in any areas of weakness. Exploring powers Explain that before the general use of calculators people used power tables to do multiplication and division. Here is part of a power table: 2 = = = = = = = = = 10 1 Use the table to work out some simple calculations. Show that the following are true: 4 2 = 2, 8 2 = 4, 3 3 = 9, 10 2 = 5, 2 5 = 10. Discuss how the table can be used to show that 5 4 = 20. This column is for schools to note their own resources, e.g. textbooks, worksheets. 117 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

6 Objectives Possible teaching activities Notes School resources Linking powers to geometric sequences Suppose that a colony of bacteria doubles every hour, starting with 100 bacteria. Ask students to work out how many hundreds of bacteria there are each hour for the next 12 hours. Discuss with them how this links to the powers of 2: 2 0, 2 1, 2 2, 2 3, 2 4, 2 5, Ask students to write a formula for the number of bacteria after n hours. Set students questions such as: How long does it take for the bacteria to increase from 3200 to ? What does the 5 show in = 2 5 in the context of the problem? The bacteria are counteracted by an antibiotic that halves the number of bacteria every hour. If this antibiotic is introduced when there are bacteria, work out how many bacteria there are over the following 12 hours. Practice Give further examples of problems involving geometric sequences for students to explore. 2 hours Proportional reasoning Solve a range of problems using the multiplicative nature of proportional reasoning. Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary. Proportional reasoning problems Ask students to solve a range of problems using multiplicative methods for proportional reasoning. For example: In the diagram, the two triangles are similar. Calculate the length a. In a pie chart showing a family s weekly expenditure, the angle representing entertainment is 65. If the total weekly expenditure was QR 1380, how much was spent on entertainment? In a scale model of a new city development, the footprint of the development is 1 50 th of the actual footprint on the ground. What is the ratio of the volume of the scale model to the volume of the full-size development? On a map with scale 1 : a lake has an area of 13 cm 2. What is the area of the same lake on a map with scale 1 : ? A solid shape has surface area 200 cm 2 and volume 5500 cm 3. A similar larger shape has surface area 350 cm 2. Both shapes are made of a material with density 2500 kg/m 3. Find the mass of the larger shape. Best buy comparisons Ask students to collect price data for the same item in different shops and for different brands and different quantities. Ask them to work in groups to decide which shop, brand and quantity offers best value for money. As they work, make sure that they are using their calculators efficiently. Get the groups to give short presentations to the whole class on what they have found out. Data could be collected by students either for homework or by using the Internet. 118 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

7 Objectives Possible teaching activities Notes School resources Converting units Ask students to research different units and to make conversion tables for converting between them for different quantities; e.g. the Romans measured lengths as follows: Roman measure Digitus Palmus Cubitus Passus Mille passus Metric equivalent 18.5 mm 74.0 mm mm 1.48 m m Useful websites Roman units: romantables.8m.com Units of measurement: 2 hours Fractions, decimals and percentages Perform harder percentage calculations, including taking a percentage of a percentage, inverse percentage and compound interest problems. Convert any recurring decimal to an exact fraction. Can you think of where some of these measurements might have originated? Do modern-day body measurements match? Practice If needed, provide similar exercises to convert imperial units to metric units. Percentage problems Challenge students to work on more complex percentage questions where they need to decide how to apply their knowledge and understanding of percentages, select the correct information from that presented and relate the calculated answer back to the context of the original problem. For example: 65.4% of the population of Qatar is male. 18.9% of the male population and 34% of the female population are under the age of 15. What percentage of the total population is under 15? A shop marks up the profit on an item in order to make a profit of 45%. In a sale, a discount is made of 15% off the marked price. If the item is sold in the sale, what percentage profit will the shop make on it? Annual equivalent percentage rates Ask students to find on the Internet rates for loans given in different forms, and use these to gain a sound understanding of how rates compare and the different ways of writing them. How much will the monthly repayments be on a car loan of QR paid back over five years? Over the period of the loan, how much of the money paid back will be interest and how much capital? What is the best rate you can find for a car loan? How much interest can you get for a deposit of QR ? In how many different ways can you express this (e.g. as a monthly rate, a quarterly rate or an annual rate)? Fractions and decimals If the denominator of a fraction is 2 or 5, what can you say about the equivalent decimal? Can you find any other rules for different denominators? Choose a number less than 10. Divide it by each prime number in turn up to 13. After how many places of decimals does each begin to recur? Can you find any rules? A useful website is Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

8 Assessment Unit 11F.1 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Calculate mentally the value of In this question, you should not use a calculator. a. An elastic band is fixed on four pins on a pinboard, as shown in the first diagram. Show that the total length of the band in this position is 14 2 units. b. What is the length of the band in the new position shown? Write your answer in its simplest form using roots. Use standard form to estimate the value of π 2. 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 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 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. The Earth completes its orbit around the Sun in 365 days. The Earth is 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. QR 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 a 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? Explain why 0.12 = Qatar mathematics scheme of work Grade 11 foundation Unit 11F.1 Number Education Institute 2005

9 GRADE 11F: Geometry 1 Proof UNIT 11F.2 3 hours About this unit This is the first of two units on geometry for Grade 11 foundation. It builds on the work on geometry in the Grade 10 foundation units. 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 10 foundation or Grade 12 foundation. 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 angles at a point, angles on a straight line, and alternate and corresponding angles to establish the congruency of two triangles. Expectations By the end of the unit, students will continue to use their knowledge of geometry and trigonometry to solve practical and theoretical problems relating to shape and space. They will use ICT to explore geometry. Students who progress further will explore aspects of geometry including geometric patterns, similarity and congruence, constructions, plans and elevations using ICT. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector dynamic geometry system (DGS) such as: Cabri Geometrie ( Geometer s sketchpad ( computers with Internet access and dynamic geometry software for students graphics or scientific calculators for students sharp pencil, straight edge and compasses for each student Key vocabulary and technical terms Students should understand, use and spell correctly: proof, conjecture, axiom, equidistant, mid-point, bisector, vertex, perpendicular, parallel, intersection, incentre, inscribed circle, tangent 121 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

10 Standards for the unit Unit 11F.2 3 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 3 hours Proof 10F.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. 11F.6.1 Use dynamic geometry systems to conjecture results and to explore geometric proof. 12F.6.1 Use ICT to investigate a range of geometrical situations, including: the generation of geometric patterns, including Islamic patterns; similarity and congruence; constructions; plans and elevations. 122 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

11 Activities Unit 11F.2 Objectives Possible teaching activities Notes School resources 3 hours Proof Use dynamic geometry systems to conjecture results and to explore geometric proof. Introductory activity Ask students to work individually to prove that the angles in a triangle add up to 180. Then ask them to work in pairs and to convince each other that they have proved this result. Each pair should then join up with another pair and decide which of their proofs is the best. The best proof is then presented to the class. Discuss with the class: Which proof was the best? How would you define best in this context? Which proof convinced you the most? What makes a good proof? Are there different proofs that are equally valid? Help students to understand the difference between a proof and a demonstration, and that there are different ways of proving things, depending upon the starting axioms. Many students will suggest a demonstration for example, by drawing triangles and measuring the angles or by tearing the corners off a triangle and fitting them together rather than a proof. They will benefit from seeing several proofs, including those below, based on a tessellation and on the alternate angle theorem. This column is for schools to note their own resources, e.g. textbooks, worksheets. 123 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

12 Objectives Possible teaching activities Notes School resources Using a dynamic geometry system (DGS) Ask students to create a quadrilateral using DGS, then to construct the mid-points of all four sides and connect them in order. Ask students to drag some of the vertices of the original quadrilateral before answering the following. Describe the quadrilateral in the centre. When you drag the vertices of the original quadrilateral, what stayed the same and what changed? How might you test to see if you are right about this conjecture? What measurements could you take? How could you change your original quadrilateral to make the inside quadrilateral a square? How do you know it is a square? Dynamic geometry software offers students the opportunity to explore and conjecture and to convince themselves by experiments in preparation for more formal methods of proof. Investigating the centres of triangles In this investigation students should aim to prove the following: The perpendicular distances from the incentre to each of the three sides of a triangle are equal. Ask students to draw a triangle that appears to be equilateral on a piece of paper. Then ask them to visualise the three interior angle bisectors of the triangle. Where do the first two cross? Where does the third bisector cross the first two? What happens if the triangle is changed to an isosceles triangle or a scalene triangle? Ask students to construct a triangle and its angle bisectors using DGS and then to check their predictions by dragging a vertex in their sketch. The point of intersection of the three angle bisectors is called the incentre of the triangle. Now ask students to think about the distances from the incentre to each of the three sides along the angle bisectors and to imagine how the distances change as a vertex of the triangle changes. They can then check by dragging in their diagram. Ask students to add the perpendicular bisectors of the sides of the triangle to their diagram and then to manipulate their diagram so that the perpendiculars coincide with the angle bisectors. What type of triangle is this? Can you prove that the perpendicular distances from the incentre to each of the three sides of any triangle are equal? Ask students to inscribe a circle in the triangle so that it stays inscribed as the triangle is dragged, to see why this centre is called the incentre. Here students are encouraged to visualise the problem before carrying out the construction using DGS and exploring it by dragging vertices. This problem has a number of applications in terms of locating a place equidistant from various other locations. 124 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

13 Objectives Possible teaching activities Notes School resources Developing proofs Give students a selection of results to prove. Proof 1 In the diagram, A is the mid-point of side XY and AB is parallel to YZ. Prove that B is the mid-point of side XZ and that AB is half of YZ. Discuss with students: What is your starting point? What axioms have you started from? What information are you given? Is all the information marked on the diagram? What new information can you deduce from the given information? For example, what do you know about parallel lines and angles? Mark the new information on the diagram. Does the information on the diagram fulfil the conditions for a theorem to be applied? Proof 2 In the diagram the circle is inscribed in the quadrilateral PQRS. Prove that PQ + SR = PS + QR. Proof 3 In the diagram A is the point at which the tangent T touches the circle. Prove that ACB = BAT. Practice Provide some further examples of problems for students to solve using geometric proof. 125 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

14 Objectives Possible teaching activities Notes School resources Awkward areas: part 1 Ask students to explore the areas of triangles inscribed within polygons. Students working on this task need to have prior knowledge of the area of a triangle, sums of the interior and exterior angles of polygons and trigonometric relationships. This task is designed to be worked on collaboratively by students working in pairs or small groups. Make resource sheets as on the right and on the next page. Each group will need paper, pen, pencil and ruler and a scientific or graphics calculator, and access to DGS. Explain to students that they will be exploring what happens to the area of triangles inscribed in different polygons as one of the vertices moves around the perimeter of the polygon. Use discussion to help students to explore the problem. For example, if students want to produce scale drawings to measure the lengths of sides of triangles within polygons, discuss issues of accuracy and encourage them to calculate the areas instead. Encourage confident writers to explain their thinking on paper. Note that in this task the side of the polygon has been fixed at 10. Choosing a fixed value enables students to concentrate in the early stages of the task on finding numerical values or expressions for areas, without consideration of another variable. However, you may prefer to replace 10 with a variable such as k. Students can also explore the problem using DGS. Resource sheet 1 ABCD is a square of side length 10 units. Point Y can move on the perimeter of the square. Triangle AYB is shaded. When Y is on side DC, what area is shaded? As Y moves on the perimeter of the square from A to D to C to B, the area that is shaded changes. Show on a graph how the area changes. 126 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

15 Objectives Possible teaching activities Notes School resources Awkward areas: part 2 Students are unlikely to complete both parts of the task in one lesson. Revisiting the task after a break allows time for reflection and further extension. Adapt the task for the whole class to work with the ideas at an appropriate level. For example, some students may consider only the square (Awkward areas: part 1). Others, whose trigonometric skills are still to be consolidated, may also work with the triangle and hexagon, or consider only the areas of each of the first triangles formed in each polygon, i.e. the triangle made by joining side AB to an adjacent vertex. Resource sheet 2 Below are some regular polygons, each of side length 10 units. Y can move on the perimeter of each polygon. Draw graphs to show how the area that is shaded changes as Y moves on the perimeter of each polygon. Imagine Y moving on the perimeter of a regular octagon that has one side AB. Without calculating, can you predict what the graph of area AYB would look like? 127 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

16 Assessment Unit 11F.2 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. In the diagram below, the altitudes BN and CM of the triangle ABC intersect at S. MSB is 40 and SBC is 20. Prove that triangle ABC is an isosceles triangle. TIMSS Grade 12 Construct a rhombus using DGS. How can you be sure that it is a rhombus and that it does not deform when its vertices are dragged? Choose one of the proofs you have developed during this unit. Write it out step by step as clearly as you can. Cut out each step and shuffle them. Give the pieces to another student to see if they can reassemble the steps of your proof in the correct order. 128 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.2 Geometry 1 Education Institute 2005

17 GRADE 11F: Algebra 1 Proportionality and quadratic functions UNIT 11F.3 7 hours About this unit This is the second of six units on number and algebra for Grade 11 foundation. It is the first in a series of four units on algebra. It builds on the work on algebra in Grade 10. 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 10 foundation or Grade 12 foundation. 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 work with straight line graphs of the form y = kx, knowing that the constant of proportionality, k, is the gradient of the line. They should be able to plot the graphs of quadratic functions of the form y = ax 2 + c and to identify intercepts with the axes, the axis of symmetry and the coordinates of the maximum or minimum point. Expectations By the end of the unit, students will solve problems where one variable varies in proportion to the square of the other. They will recognise when quadratic functions are increasing, decreasing or stationary. They will use a range of strategies to solve problems. Students who progress further will appreciate a wide range of algebraic applications in the real world. They will use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, sine and cosine functions, and the modulus function and other simple non-standard functions. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograph (see Graphmatica (free from www8.pair.com/ksoft) computers with Internet access and graph plotting software for students graphics calculators for students string, heavy object to act as a weight, clamp stand and stopwatch for each group of students graph paper Key vocabulary and technical terms Students should understand, use and spell correctly: proportional, y = kx 2, parabola second-order polynomial, quadratic function, y = ax 2 + bx + c, intercept, axis of symmetry, maximum, minimum, increasing, decreasing, stationary, constant, variable 129 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

18 Standards for the unit Unit 11F.3 7 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 3 hours Proportionality 4 hours Graphs of quadratic functions 10F F.5.7 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. Know that if two coordinate variables are connected by a straight line graph that passes through the origin of coordinates, then each coordinate variable is proportional to the other; use relevant information to find k. 11F.5.4 Translate the statement y is proportional to x 2 into the symbolism y x 2 and into the equation y = kx 2 ; know that the graph of this equation is a parabola through the origin. 10F.5.14 Recognise a simple second-order polynomial in one variable, y = ax 2 + c, as a quadratic function; plot graphs of such functions, and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point. 11F.5.9 Recognise a second-order polynomial in one variable, y = ax 2 + bx + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas) and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when such functions are increasing, when they are decreasing and when they are stationary. 12F.5.2 Use physical contexts to plot and interpret: graphs of linear, quadratic and cubic functions; graphs of the reciprocal function y = k/x (x 0); graphs of the sine and cosine functions; graphs of the modulus function and a range of simple non-standard functions. 11F.1.5 Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary. 130 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

19 Activities Unit 11F.3 Objectives Possible teaching activities Notes School resources 3 hours Proportionality Translate the statement y is proportional to x 2 into the symbolism y x 2 and into the equation y = kx 2 ; know that the graph of this equation is a parabola through the origin. Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary. Investigating relationships Ask students to investigate the relationships in sets of data such as those below: x y x y x y The first table shows a relationship between x and y that is directly proportional, which students have worked with previously. The second and third tables do not follow this pattern. What patterns can you see in these tables? If you order the pairs of points in one of the tables, how fast do the y numbers grow? What could you do to find out more about the relationships between x and y? Encourage students to consider the order of the data points and whether looking at differences would help, and to draw a graph to gain a picture of the relationship. This approach should lead students to consider a quadratic and therefore should hint of a way of expressing the relationship. Pendulum experiments Ask students to work in small groups to carry out experiments with a pendulum to explore a practical example where one variable is proportional to the square of the other. Tie a heavy object to one end of a piece of string and attach the other end of the string to a clamp stand or beam. Allow the weight to swing backwards and forwards. Use a stopwatch to find the time for 10 complete swings backwards and forwards and hence the time for one swing. Ask students: What variables could you change? How do you think the variables are related? Ask students to choose which of the weight, length of string or angle of release they wish to change and to design an experiment to find out the relationship between the time of the swing and their chosen variable. They should plot a graph of their results and present them to the rest of the class. What kind of relationship does the graph show? Students should find that the angle of release and the weight make no difference, but that the square of the time is proportional to the length of the string. This column is for schools to note their own resources, e.g. textbooks, worksheets. 131 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

20 Objectives Possible teaching activities Notes School resources Braking distances Ask students to use the Internet to find out the stopping distances for cars at different speeds. Ask them to plot the data using a computer or graphics calculator and to find an approximate relationship between the two variables. How accurate do you think your model is? How could you improve your model? For the example shown, the quadratic regression function given by a calculator is d = s s. Is this sensible? What happens, for example, at low speeds with this function? A web search using braking distances will give a number of suitable websites, such as Applications of square proportional relationships Ask students what other relationships they have encountered where one variable is proportional to the square of the other. Examples might include: surface area and length; scale factor and surface area; height and distance to the horizon; height of a ball thrown up in the air and speed at which it is thrown. Give students opportunities to solve problems using these kinds of contexts. Look for suitable examples in your school s textbooks. 132 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

21 Objectives Possible teaching activities Notes School resources 4 hours Graphs of quadratic functions Graphs of quadratic Discuss a range of graphs of quadratic functions with the class. For example: functions The diagram shows the graph with equation y = x 2. Recognise a secondorder polynomial in On the same axes, sketch the graph with equation y = 2x 2. one variable, y = ax 2 + bx + c, as a Curve B is the translation one unit up the y-axis of y = x 2. quadratic function; plot What is the equation of curve B? graphs of such functions (recognising that these are all parabolas) and pick out the intercepts with the coordinate axes, the axis of symmetry and The diagram shows a sketch of the curve y = 16 x 2. the coordinates of the What are the coordinates of points A, B and C? maximum or minimum point; understand when such functions are increasing, when they are decreasing and when they are stationary. Curve A is the reflection in the x-axis of y = x 2. What is the equation of curve A? The curve y = 16 x 2 is reflected in the line y = 12. Point B 1 is the reflection of B. What are the coordinates of B 1? What is the equation of the new curve? 133 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

22 Objectives Possible teaching activities Notes School resources Graphing quadratics Ask students to use a graphics calculator or a graph plotting program to plot the graphs of quadratic equations in the form y = ax 2 + bx + c. What effects do a, b and c have on the curve? Try changing only one variable at a time. What general shape is the curve? What happens if a is negative? What happens if c is negative? Practice Ask students to use a graphics calculator or a graph plotting program to complete the table for each of the following functions. y = x 2 y = x y = 2x 2 y = x 2 3 y = x Function x-intercept y-intercept Coordinates of maximum/minimum Axis of symmetry Do you notice any patterns? Can you think of any quick ways to find this information out directly from the equation without plotting it? Give each student two pieces of paper, each with a set of axes labelled and marked on it. Ask each student to write a simple quadratic equation, draw the table of values and plot the graph. This should be kept hidden from other students. Students should then pair up and take turns to describe their graph to the other student in terms of its intercepts, maximum or minimum, the change in gradient and the axis of symmetry. The second student should try to sketch the graph on a blank set of axes. The aim is to give good enough instructions for the other student to reproduce the graph without having seen it. Ask students to predict the shapes of the graphs of: y = 2x 2 y = 2x y = 2x 2 + 6x + 3 Ask different students to share their predictions with the class and to justify them. Ask students to use a graphics calculator or a graph plotting program to plot all three functions to check their predictions. They should then use a calculator or software to create tables of values for the functions for a range of values of x and compare these with their graphs and predictions. What effects do the additional parts of the function have on the graph? 134 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

23 Objectives Possible teaching activities Notes School resources Applications of quadratic functions 1 Discuss some examples of parabolas in real-world applications. For example: Ali throws a ball to Bader standing 20 m away. The ball is thrown and caught at a height of 2.0 m above the ground. The ball follows the curve with equation y = 6 + c(10 x) 2, where c is a constant. Calculate the value of c by substituting x = 0, y = 2 into the equation. The ball is thrown again but this time hits the ground before reaching Bader. This time the ball follows the curve with equation y = 0.1(x 2 6x 16). Calculate the height above the ground at which the ball left Ali s hand. Applications of quadratic functions 2 The graph shows the rate at which cars left a car park between 1700 and 1800 hours. The lowest rate was 10 cars per minute. The highest rate was 40 cars per minute. y = ax 2 + bx + c is the relationship between y, the number of cars leaving per minute, and x, the number of minutes after 1700 hours. Explain how you can work out from the graph that the value of c is 10. Use the graph to form equations to work out the values of a and b in the equation y = ax 2 + bx + c. 135 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

24 Assessment Unit 11F.3 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. A body falling from rest under the force of gravity falls a distance s metres in time t seconds, where s = 4.9t 2. Find the distance fallen after 5 seconds. How long does it take the body to fall 30 metres? Discuss how to plot a linear graph s = 4.9z, by defining the variable z = t 2. A child s toy consists of a set of nesting bowls. One of the bowls is 5 cm high and has a surface area of 20 cm 2. Find the surface area of one of the other bowls which has height 8 cm. The length of a pendulum l is directly proportional to the square of the time t for a complete swing of the pendulum. Given that the length of a pendulum which has time for a complete swing of 1.5 seconds is cm, what is the length of the pendulum with a swing that takes 1.8 seconds? The distance to the horizon can be calculated as 2.12 times the square root of the height of your eye measured in metres. This gives the distance to the horizon in nautical miles. Suppose you were in a fishing boat and 3.1 m above sea level. How far is the horizon? A tall tower in a port is 102 m high. If you are in the same fishing boat as before, how far away can you see the tower? 136 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.3 Algebra 1 Education Institute 2005

25 GRADE 11F: Geometry and measures 1 Solving problems in triangles UNIT 11F.4 6 hours About this unit This is the first of four units on geometry and measures for Grade 11 foundation. It builds on the work on geometry and measures in Grade 10 foundation and on Unit 11F.2, Geometry 1. 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 10 foundation or Grade 12 foundation. 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 be able to use Pythagoras theorem to find the distance between two points and to set up the equation of a circle. They should already know the trigonometric ratios for sin θ, cos θ and tan θ and be able to use them and Pythagoras theorem to solve right-angled triangles. Expectations By the end of the unit, students will continue to use their knowledge of geometry and trigonometry to solve practical and theoretical problems relating to shape and space. They will solve right-angled triangles in two and three dimensions using the standard trigonometric ratios. They will know and use the sine and cosine rules, and calculate the area of a triangle using 1 2 ab sin C. They will break down problems into smaller tasks. They will work to expected degrees of accuracy, and know when an exact solution is appropriate. Students who progress further will use their knowledge of geometry and trigonometry to solve more complex practical and theoretical problems relating to shape and space. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph paper sharp pencil, ruler, compasses and scientific calculator for each student Key vocabulary and technical terms Students should understand, use and spell correctly: angle of inclination, angle of declination, sine, cosine, tangent, trigonometry Pythagoras theorem, sine rule, cosine rule equilateral, polygon, regular, icosahedron, pentagon, perpendicular 137 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

26 Standards for the unit Unit 11F.4 6 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 2 hours Solving problems in right-angled triangles 4 hours Solving problems in any triangle 10F F.6.7 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; use these ratios to find the angles of a right-angled triangle given two sides, or to find the remaining sides given one side and one angle. Use Pythagoras theorem to find the distance between two points and to solve right-angled triangles; set up the Cartesian equation of a circle of radius r, centred at the origin of an xy-coordinate system. 11F F F.6.3 Solve right-angled triangles using the standard trigonometric ratios, including tan θ = sin θ / cos θ, and/or 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. Know and use the sine rule and the cosine rule to solve triangles. 11F.6.5 Calculate the area of a triangle using 1 2 ab sin C. 11F.6.4 Solve triangle problems in two and three dimensions. 11F.1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate. 11F.1.4 Break down complex problems into smaller tasks. 138 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

27 Activities Unit 11F.4 Objectives Possible teaching activities Notes School resources 2 hours Solving problems in rightangled triangles Solve right-angled triangles using the standard trigonometric ratios, including tan θ = sin θ / cos θ, and/or Pythagoras theorem. Derive and recall the exact values for the sine, cosine and tangents of 0, 30, 45, 60, 90 and use these in relevant calculations. Work to expected degrees of accuracy, and know when an exact solution is appropriate. Break down complex problems into smaller tasks. Sine, cosine, tangent Display diagrams of an equilateral triangle with side length 2 units and of a right-angled isosceles triangle where the two equal sides are each of length 1 unit. Initially, give students blank diagrams to work with and ask them to add the information they know such as the angles and other side lengths using Pythagoras theorem. Now ask them to use the diagrams to work out the exact values of the sine, cosine and tangent for 30, 60 and 45. What connections do you notice between the values? How will you remember them? What are the advantages of using these common values in this form? Ask students to imagine a triangle where one angle is very small. Ask them to imagine what happens as the angle gets smaller and smaller and approaches zero. What are the values of the sine, cosine and tangent of 0? Can you use the same diagram to add the sine, cosine and tangent of 90 to your list? Ask students to find as many relationships and connections as they can between sine, cosine and tangent. They should, for example, be able to notice that the graph of cosine is that of sine with a shift so that cos θ = sin (90 θ). They should also be able to find that tan θ = sin θ / cos θ either by looking at the formulae or from tables of values. Problem solving Students should develop their proficiency in using trigonometry and Pythagoras theorem to solve problems using right-angled triangles. Give students opportunities to solve problems where they need to choose which pieces of information to use and which formulae to apply. Students will have created suitable tables and graphs for this activity in Unit 10F.12. This column is for schools to note their own resources, e.g. textbooks, worksheets. Justifying which triangles are possible Give students a selection of triangles with given dimensions (such as those on the right) and ask them to justify through reasoning and calculation (not construction) whether they are possible. Courtyard ABCD is a rectangular courtyard containing a well at point P. The well is 5 m from corner A of the courtyard, 10 m from corner B and 11 m from corner C. How far is it from corner D? Hint: the dotted lines divide the shape into right-angled triangles so that Pythagoras theorem can be applied. Source: Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

28 Objectives Possible teaching activities Notes School resources Car parks A standard car park space is 2.4 m wide and 4.8 m long. In a long, narrow car park, the spaces are often angled to make the most efficient use of space and easier access for car drivers. Find the width of the area needed for car parking for different angled bays. Which do you think would make the best car park space and why? 4 hours Solving problems in any triangle Know and use the sine rule and the cosine rule to solve triangles. Calculate the area of a triangle using 1 2 ab sin C. Solve triangle problems in two and three dimensions. Work to expected degrees of accuracy, and know when an exact solution is appropriate. Break down complex problems into smaller tasks. Investigation Ask students to draw six non-right-angled triangles on a piece of paper. Then ask them to measure the sides and angles and to fill in a table with the following headings. A B C a b c What do you notice about the final three columns? Sine rule Discuss with students the proof that for, any triangle ABC: a sin A = b sin B = c sin C (sine rule) In ACD, sin A = h b. In BCD, sin B = h a. Eliminating h gives b sin A = a sin B, or a sin A = b sin B. a sin A b sin B c sin C Give pairs of students diagrams with the perpendicular dropped from A, B or C. Ask them to find the sine of two angles and see what they can deduce. Give students the proof with each part on a separate slip of paper. Ask them to work in pairs to put the pieces in order to create the proof. Discuss with students when the sine rule would be useful and which pieces of information are needed in order to use it to solve a problem. Cosine rule Discuss with students the proof of the cosine rule that: a 2 = b 2 + c 2 2 bc cos A In ACD, b 2 = x 2 + h 2 and cos A = x b In BCD, a 2 = h 2 + (c x) 2 a 2 = h 2 + c 2 2cx + x 2 Substituting for h gives a 2 = b 2 x 2 + c 2 2cx + x 2 a 2 = b 2 + c 2 2cx Substituting for x gives a 2 = b 2 + c 2 2bc cos A Use similar activities to those above. Practice Give students a variety of triangles to solve by applying the sine or cosine rule. Extension: Ask students to draw some circles and then to draw a triangle in each with vertices on the circumference. What do they notice about the ratios a sin A, b sin B and c sin C compared with the radius of the circle? Dropping the perpendicular from a different vertex gives the third part of the rule connecting side c and sin C. There is also the case to consider where the perpendicular h lies outside the triangle. Dropping the perpendicular from a different vertex gives the rule using different combinations of sides; rearranging gives an angle in terms of three sides. There is also the case to consider where the perpendicular h lies outside the triangle. 140 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

29 Objectives Possible teaching activities Notes School resources Area of a triangle Remind students that the area of a triangle is half the base multiplied by the perpendicular height, or 1 2 bh in ABC on the right. In the right-angled triangle ABX, h = c sin A, so the area of ABC is 1 2 bc sin A. Show students that the formula still applies when the perpendicular height h falls outside the triangle. Problem solving Ask students to solve a range of problems where they need to choose which formulae to use and where they solve problems in a range of contexts presented using different phrasing, for example angle of inclination and angle of declination or depression. Give them opportunities to consider whether both of the solutions with the sine rule applies or only one does. From two points X and Y on level ground, the angles of inclination to the top, P, of a minaret are 36.1 and 51.9 (see diagram 1). The distance between X and Y is 41 m. Find angle XPY, the length XO and the height of the minaret. Find the surface area of an icosahedron with edge length 4 cm. Find the area of a field in the shape of a regular pentagon with side length 50 m. Compare the areas of a regular pentagon and the pentagonal star drawn inside it (see diagram 2). Is the shaded area equal to the unshaded area, or is one a little larger than the other? Draw a diagram with a road and a field in the shape of an irregular polygon next to it (see diagram 3). Mark the lengths of the fences on your diagram and find the area of your field by dividing it into triangles. How many more lengths do you need to know? What is the most efficient method of finding the area? Write out your solution step by step and swap it with someone else for them to check. Diagram 1 Diagram 2 Diagram 3 Squares in triangles Pose this set of problems. What is the largest square you can draw inside an equilateral triangle of side length 10 cm? What is the best way to position it? Are there any other ways of positioning the same size of square inside the triangle? How do you know you have the largest square? 141 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

30 Assessment Unit 11F.4 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. A ship sails 50 kilometres in a direction 032 and then 29 kilometres in a direction 315. How far is the ship from its starting point? What is its bearing from its starting point? A helicopter at airfield A received a distress call from a boat. The position of the boat, B, was given as 147 km from the airfield, on a bearing of 072. A man on the boat is flown to hospital. Calculate the distance the helicopter travelled from the boat to the hospital at H. Show that Pythagoras theorem is a special case of the cosine rule. Calculate the exact area of an equilateral triangle with sides of length 6 cm. A triangle has its three angles in the ratio 2 : 3 : 4. Find to two significant figures the ratio of the lengths of its sides. The Great Pyramid of Cheops in Egypt is built on a square base with side 230 metres. Each face of the pyramid is at 52 to the horizontal. Calculate the height of the pyramid. Calculate the inclination of an edge of the pyramid to the horizontal. The Great Pyramid of Cheops at Giza Source: Qatar mathematics scheme of work Grade 11 foundation Unit 11F.4 Geometry and measures 1 Education Institute 2005

31 GRADE 11F: Statistics 1 Sampling, collecting and representing data UNIT 11F.5 6 hours About this unit This is the first of two units on statistics for Grade 11 foundation. It builds on the work on statistics in Grade 10. 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 10 foundation or Grade 12 foundation. 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 know that data collected from samples can be categorical or quantitative, and that quantitative data may be discrete or continuous. They should be able to plot simple histograms in which the height of the bars is proportional to the frequency of each class interval. They should understand the idea of a random variable. Expectations By the end of the unit, students will plan questionnaires and surveys to collect meaningful primary data from samples. They will know the importance of representative samples, and will be able to locate sources of bias. They will collect data from secondary sources, including the Internet, and ask and answer questions related to the data. They will construct histograms and cumulative frequency distributions, grouping continuous data when necessary. They will draw stem-and-leaf diagrams and box-andwhisker plots and use them to present their findings. Students who progress further will plan surveys and design questionnaires to collect primary data to effectively represent the population as a whole. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector statistics software such as Autograph (see computers with Internet access and statistics software for students graphics calculators for students graph paper Key vocabulary and technical terms Students should understand, use and spell correctly: histogram, frequency, (cumulative) frequency distribution, frequency density, relative frequency, relative frequency distribution, range, percentile, interquartile range, semi-interquartile range, average, mean, mode, modal class, modal frequency, variable stem-and-leaf diagram, stem plot, box-and-whisker plot, box plot random sample, variation, population, sample, representative sample, questionnaire, survey, experiment, primary data, secondary data, hypothesis, bias 143 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

32 Standards for the unit Unit 11F.5 6 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F foundation standards EXTENSION STANDARDS Grade 12F standards 3 hours Designing a statistical enquiry 3 hours Representing data 10F.8.1 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); 11F.8.1 Know that: it is important to choose representative samples; in a random sample there are chance variations; in a biased sample there are systematic differences between the sample and the population from which it is drawn; and locate obvious sources of bias within a sample. understand the concept of a random variable. 11F.8.2 Plan surveys and design questionnaires to collect meaningful primary data from samples in order to test hypotheses about or estimate characteristics of the population as a whole; formulate problems using secondary data from published sources, including the Internet. 12F.10.2 Plan surveys and design questionnaires to collect meaningful primary data from samples (including data collected in other subjects, such as science, geography or history) in order to make estimates of, or test hypotheses about, quantities or attributes characteristic of the population as a whole. 11F.1.7 Explain their reasoning, both orally and in writing. 11F.1.11 Conjecture alternative possibilities with What if? and What if not? questions. 10F.8.3 Plot simple histograms in which the height of the bars is proportional to the frequency of each class interval and use related vocabulary, including frequency, range and mode, modal class and modal frequency. 11F F.8.5 Construct (relative frequency) histograms and plot cumulative frequency distributions, grouping continuous data when necessary. Draw stem-and-leaf diagrams and box-and-whisker plots and use them in presenting conclusions. 144 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

33 Activities Unit 11F.5 Objectives Possible teaching activities Notes School resources 3 hours Designing a statistical enquiry Know that: it is important to choose representative samples; in a random sample there are chance variations; in a biased sample there are systematic differences between the sample and the population from which it is drawn; and locate obvious sources of bias within a sample. Plan surveys and design questionnaires to collect meaningful primary data from samples in order to test hypotheses about or estimate characteristics of the population as a whole; formulate problems using secondary data from published sources, including the Internet. Explain their reasoning, both orally and in writing. Conjecture alternative possibilities with What if? and What if not? questions. Sampling activity Prepare a 10 by 10 grid, each cell having between one and ten dots, and give a copy to each student. First ask students to guess how many dots there are on the whole grid, then ask them to suggest ways of making a more accurate estimate without counting all the dots. Suggest students use a random number generator on a calculator to produce five random numbers between 1 and 100 to select five cells. They can find an estimate for the number of dots on the grid by adding the numbers of dots in the five cells and multiplying by 20. What variation in estimates is there across the class? Do any students have the same numbers of dots for their sample? Can you use the set of estimates from the class to make a better estimate of the total for the grid? How could you make a more accurate estimate of the total for the grid? Sources of bias Ask students to identify possible sources of bias in some examples. They should then identify the population to be sampled and come up with a way of avoiding this particular source of bias, giving their reasons. Possible examples are: The sample for a survey about household expenditure uses addresses as a source of respondents. A researcher carrying out a survey on people s views on traffic congestion asks the views of people entering a coffee shop between 10 and 11 o clock on a Tuesday morning. A student wants to know what leisure facilities are needed by people his own age, so he asks ten of his friends. What makes a good questionnaire? Ask students to write some advice for other students about how to design a good questionnaire. They should include some examples of what not to do. What are the pitfalls? What things are most important? How can you encourage people to be truthful and accurate in answering questions? How can you make sure that the questions are easy to answer? What is a good way of finding out the age of a respondent? This column is for schools to note their own resources, e.g. textbooks, worksheets. 145 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

34 Objectives Possible teaching activities Notes School resources Planning a project Ask students to form some hypotheses and then to work in pairs to design a questionnaire to test them. Students could choose their own areas to investigate or use one of the ideas below as a starting point. Whether people would be in favour of making a car-free zone in the shopping area of your city What students think of the meals provided in the school cafeteria What young people like to watch on television How people spend their leisure time What young people do in their summer holidays Ask pairs to swap their questionnaire with another pair and to constructively criticise the other pair s work. During this unit students should have the opportunity to plan a more substantial and open-ended project. Unit 11F.11, in which students carry out a statistical project, includes further ideas for suitable projects. 3 hours Representing data Construct (relative frequency) histograms and plot cumulative frequency distributions, grouping continuous data when necessary. Draw stem-and-leaf diagrams and box-andwhisker plots and use them in presenting conclusions. Internet research Ask students to use the Internet to find out how market research companies, media research agencies and polling organisations find out the views of others. Some possible questions are: How are television audiences measured? How are election results forecast? How do companies ensure that their samples are representative of the population as a whole? Heights of students This box plot shows the heights of 100 school-age students. Get students to use the trace facility on a graphics calculator to find the maximum, minimum, upper and lower quartiles and median for both sets of data. The upper box plot gives the data for the girls and the lower for the boys. Ask students to write a paragraph to describe and compare the distributions in as much detail as possible, and explain their conclusions. Extend the discussion by asking some What if? questions, such as: What if there were 1000 school age students in the sample? Would you expect the box plots to be different? Why? Students should already be familiar with and have constructed on paper the types of graphs, charts and tables that they generate with ICT. Students should have opportunities to use technology to produce graphical representations of data, including via the Internet, computer software and graphics calculators. Websites such as activities/tools.html provide resources for producing statistical diagrams that can use your own data. 146 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

35 Objectives Possible teaching activities Notes School resources Travel times The box plots on the right, produced with Autograph, show the time taken (in minutes) for a random sample of students to travel to school in the UK. Which do you think is the plot of travel time for students in Grades 5 and 6 and which is for students in Grades 11 and 12? Why? What can you say in comparing the two box plots? Do you think this is a good way of representing the data? Here is a back-to-back stem-and-leaf diagram for the same data. Grades 5 and 6 Grades 11 and Key: 5 10 and 10 5 mean 15 minutes A useful source of data is What can you tell about travel times from the diagram? Explain your reasoning. What different information is given by the two charts? Real data Real data collected by students, using their questionnaires developed in the first part of this unit or from the Internet, will provide some meaningful contexts for drawing and interpreting statistical diagrams. For example, students could examine population data in Qatar using the source Get students to draw histograms to represent these data. Ask: What can you say about the distribution of the ages of the population in Qatar? What can you say about Qatar s expected population, currently and in the near future? What differences are there between the two years? Give possible reasons to explain these differences. What differences are there between males and females? Give possible reasons to explain these differences. Looking at predicted data allows What if? questions to be asked, such as: What if Qatar s sources of natural gas ran out? What impact do you think this would have on the population distribution? The population of Qatar by age in 2000 and predicted for Age Male Female Male Female Source: Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

36 Assessment Unit 11F.5 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. An article in a newspaper claimed that 93% of us drop litter every day. Some students think the percentage of people who drop litter every day is much lower than 93%. They decide to do a survey. a. Jabor plans to ask 10 people if they drop litter every day. Give two different reasons why Jabor s method might not give very good data. b. Layla plans to go into Doha on a Sunday morning. She will stand outside a shop and record how many people walk past and how many of those drop litter. Give two different reasons why Layla s method might not give very good data. Mosa wants to investigate whether more people are born in the winter than in the summer. He plans to ask 30 students in his grade whether they were born in the winter or the summer. Discuss ways in which Mosa could improve his survey. 304 people took part in a swimming contest. They swam 1.5 km. The histogram shows the distribution of their times for the event. a. The histogram is constructed using frequency densities. The table shows the frequency densities. Complete the table to show the frequencies. Time t (minutes) Frequency density Frequency 17 t < t < t < t < b. Calculate an estimate of the mean time. c. Explain why the median time must be between 22 and 27 minutes. d. Calculate an estimate of the median time. A hospital clinic records in a stem-and-leaf diagram the number of patients seen each day Key: 12 5 means 125 patients. Find the range, the median and the mode. Draw a box-and-whisker plot for these data. 148 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.5 Statistics 1 Education Institute 2005

37 GRADE 11F: Algebra 2 Properties of functions and inequalities UNIT 11F.6 9 hours About this unit This is the third of six units on number and algebra for Grade 11 foundation. It is the second in a series of four units on algebra and builds on Unit 11F.3, Algebra 1. 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 10 foundation or Grade 12 foundation. 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 plot straight -line graphs of the form y = mx + c, and to relate the gradient and intercepts on the axes to m and c. They should be able to identify the equation of a straight line from its graph alone, or from the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line. Expectations By the end of the unit, students, through their continued study of linear, quadratic, reciprocal and other functions and their graphs, and the solution of associated equations, will appreciate a range of numerical and algebraic applications in the real world. They will solve simple problems represented by regions of linear inequality. They will recognise when functions are increasing, decreasing or stationary. They will find the tangent at a point on the graph of a function. They will aim to generalise. Students who progress further will appreciate a wide range of numerical and algebraic applications in the real world. They will use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, sine and cosine functions, the modulus function and other simple non-standard functions. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograph (see Graphmatica (free from www8.pair.com/ksoft) computers with graph plotting software for students graphics calculators for students graph paper and squared paper lesson plan 11.2 Key vocabulary and technical terms Students should understand, use and spell correctly: function, continuous, discontinuous, tangent, increasing, decreasing, stationary, piecewise function implicit form, explicit form, gradient, slope, rate of change of y with respect to x, intercept on the x- or y-axis, symmetry inequality, linear programming, region, constraint 149 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

38 Standards for the unit Unit 11F.6 9 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 11F and 12F standards 6 hours Properties of functions and their graphs 3 hours Inequalities 10F F F.5.10 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them. Know that a straight line in the explicit form y = mx + c represents a function; plot the graphs of such functions, relating the gradient of the line and intercept on the x- or y-axis 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. 11F F F F F.5.8 Use a graphics calculator to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts. 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. 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 equations in 11F.5.6; know the meanings of gradient of the line (and be familiar with alternative wordings such as slope or rate of change of y with respect to x), 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. Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions; understand simple quadratic inequalities. 11F F.5.2 Recognise when a graph represents a functional relationship between two variables and when it does not. Use physical contexts to plot and interpret: graphs of linear, quadratic and cubic functions; graphs of the reciprocal function y = k/x (x 0); graphs of the sine and cosine functions; graphs of the modulus function and a range of simple non-standard functions. 11F.1.9 Aim to generalise. 150 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

39 Activities Unit 11F.6 Objectives Possible teaching activities Notes School resources 6 hours Properties of functions and their graphs Use a graphics calculator to plot and interpret a range of functional relationships, some continuous and others discontinuous, arising in familiar contexts. 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. 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] Exploring functions 1 Use these keys of a graphical calculator: [WINDOW] to draw the axes; [Y= ] to input the function in the form y = ; [GRAPH] to draw the graph; [TRACE] to trace along the graph. Use this activity as a starting point for further exploration; for example, to: recognise that a graph of the function with equation y = 4x 1 is a set of points whose coordinates satisfy the relationship y = 4x 1; investigate the graph of y = mx + c to investigate perpendicular lines you will need to set the WINDOW to give square axes, e.g. WINDOW [ 4.7, 4.7, 1, +3.1, 3.1,.5, 1]; solve linear equations; solve pairs of simultaneous linear equations. Linear functions from polygons Give students some diagrams showing various polygons in different orientations on a grid. Ask them to write equations for the lines that form the sides of these shapes. Which of these are implicit functions and which are explicit? Are there any lines that cannot be written in explicit form? Why? Note that graphics calculators and graph plotting software will only allow entry of functions in explicit form. Practice Ask students to work in pairs. Students take it in turns to choose a function. The other student has one minute to write some equivalent equations for that function where the function has been rearranged in both explicit and implicit forms in as many different ways as possible. For example: y = 2x + 3, 2x y + 3 = 0, x = 1 2 (y 3) etc. This column is for schools to note their own resources, e.g. textbooks, worksheets. Lesson plan Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

40 Objectives Possible teaching activities Notes School resources [continued] Plot the graphs of equations in 11F.5.6; know the meaning of gradient of the line (and alternative wordings), 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. Aim to generalise. Implicit and explicit forms of linear functions Ask students to plot functions given in both explicit and implicit forms. A suitable set of functions might be: x + y = 3 y = 3x 5 x = 4 2x 4y = 9 y = 7 2y 3x = 4 Ask students to predict what the graph will look like and sketch it before plotting it. Ask them: How can you decide whether the line has positive or negative gradient in both the explicit and implicit form? When given the implicit form, how can you spot the intercept without plotting the graph? Exploring functions 2 Ask students to use a graphics calculator or graph plotting software such as Autograph to graph a range of functions and to classify them. A suitable set of functions might be: y = y = x 3 1 x 3 y = x x y = 1 2 x 3 y = x y = x Students should sketch the functions and note their distinctive features. Exploring functions 3 Give students a range of graphs of continuous and discontinuous functions printed out from a graphics calculator or graph plotting software. Ask them to reproduce the graphs using ICT and to identify the functional relationship(s) involved. Modelling with quadratic functions Ask students to solve problems such as the following. A girl throws a baseball. At the moment of the throw the ball is 2 m above the ground, which is horizontal. The baseball just goes over a wall of height 4 m that is 20 m away from the girl. When the ball hits the ground it is 30 m from the girl. Find the equation that describes the flight of the baseball. The central section of a suspension bridge across a wide expanse of water can be modelled by a quadratic equation. Given some dimensions, find the equation of the parabola formed by the hanging cable and solve some related problems. 152 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

41 Objectives Possible teaching activities Notes School resources Piecewise functions Give students sets of cards where some have graphs of piecewise-defined functions on them and others have the equations for parts of the functions. Ask students to work in pairs to match these up and to provide the domains for each equation. Suitable functions include: For example, one set might be: y = 3x + 5 for x 0 y = 3x + 5 for x 0 Tangents Ask students to draw an accurate graph of y = x 2 on a piece of A4 graph paper, making the graph as large as possible. They should then work in small groups to draw tangents for different values of x and work out the gradient of the function at that point as accurately as possible. The work from all the students in their group should be collected to form a shared table of results. Question students as follows: What do you notice about the gradient of the function? Can you see evidence for the symmetry of the function? Where is the gradient zero? Can you describe what happens to the gradient as you approach the turning point? What happens as you continue and move past it? What happens to the gradient as x gets very large? Students should now choose another function, perhaps a cubic or hyperbolic function. Before drawing and checking the gradient at particular points they should write down how they expect the gradient to change at the turning points of the function. Extension problem 1 Here is a pattern for you to experiment with. This one is composed of the graphs of 14 parabolas. You may like to use a graphics calculator or graph drawing software to do this question. The equations of three of the graphs are: y = 2(x 4) y = 2x 2 1 y = 2(x 6) 2 Find the equations of the other 11 graphs in this pattern. Extension problem 2 Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x) = 0 is Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

42 Objectives Possible teaching activities Notes School resources Extension investigation: the square root and integer functions Which of the following are always true, which are sometimes true, and which are never true? Give a proof of your assertions if you can, or some examples if not. SQR(x + 4) = SQR(x) + 4 (never true; squaring both sides, x would equal x + 8 SQR(x) + 16) INT(x + 4) = INT(x) + 4 (always true; if x = a + b, where a is a positive integer and 0 b < 1, each side equals a + 4) INT(x + 7.5) = INT(x) (never true, since the left side is integral and the right side is not) SQR(xy) = SQR(x) SQR(y) (always true; each side is positive and its square is xy) INT(xy) = INT(x) INT(y) (sometimes true; when x = a + b and y = c + d as in the second example above, the left-hand side is INT(ac +[bd + ac + bd]) and the right-hand side equals ac, which is true if and only if 0 (xy ac) < 1) SQR(x) represents the square root of x, so that SQR(1.44) = 1.2. INT(x) represents the integer part of x, so that INT(5.73) = 5. The following graph is of y = INT(3x) 3 INT(x) for values of x from 0 to 6. Further practice and investigations Get students to use a graphics calculator or graph plotting software to: solve higher order equations; solve pairs of simultaneous equations; make general statements about the number of solutions of a quadratic, cubic, equation; find the minimum or maximum of a quadratic function; investigate the graph of the function y = a(x + b) 2 + c when a, b and c take different positive and negative values. 3 hours Inequalities Graph regions of linear inequality and solve simple problems (e.g. elementary linear programming) represented by such regions; understand simple quadratic inequalities. Demonstrating inequalities using diagrams Demonstrate some inequalities using diagrams: for example, the inequality (a + b) 2 a 2 + b 2. Two rectangles, each of area ab, fit inside the two squares of areas a 2 and b 2, showing that a 2 + b 2 2ab. The inequality a 2 + b 2 + c 2 ab + bc + ca can be shown by fitting three rectangles of areas ab, bc and ca inside three squares with sides a, b and c. 154 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

43 Objectives Possible teaching activities Notes School resources Demonstrating inequalities by plotting regions Ask students to consider the function y = 2x + 3 and to plot the graph on squared paper. Discuss with students: Where are the points that have y equal to 2x + 3? Where are the points that have y less than 2x + 3? Where are the points that have y greater than 2x + 3? How can we show this on the graph? What would be a quick way to find out which side of the line we need? Encourage students to choose some points, work out the value of 2x + 3 at those points and then put a tick or a cross to show whether the inequality is true at that point in order to identify the region. Practice Provide similar examples for students to practise solving problems involving inequalities by plotting graphs. Using technology Encourage students to use a graphics calculator or suitable software package to generate graphs of inequalities and to identify regions where two or more inequalities are satisfied. Simple linear programming Ask students to work with simple applications of inequalities, including the solving of simple linear programming problems where a number of constraints are satisfied simultaneously. For example: A cake company produces two types of celebration cake. The first type takes four hours to make per batch and two hours to cook. The second type takes two hours to make per batch and two hours to cook. The maximum cooking and making time available each week is 64 hours in total. The company makes QR 48 profit on the first type of cake and QR 35 on the second type. How many of each type of cake should the company make in order to make the maximum profit? Graph to show 2x 3y 12 and x + 5y Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

44 Assessment Unit 11F.6 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Plot the graph of y = 1/x 2 for the domain set {x: x and 1 x 4}. Discuss whether the domain could be extended. Plot the curve y = x on a suitably defined domain. Discuss why the domain cannot be the set. Compare this curve with the curve of y = x 2, drawn on the same axes. Invent examples of functions with different definitions on different subdomains: for example, the cost of electricity as a function of the number of units of electricity used. 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, measured in metres. The enclosure has an area of 600 m 2. Show that the total length, L metres, of fencing is given by L = d /d. Plot this function using a graphics calculator. Find from the graph the value of d that makes L as small as possible. The shaded region is bounded by the curve y = x 2 and the line y = 2. Circle two inequalities which together fully describe the shaded region. y < x 2 x < 0 y < 2 y > 0 y > x 2 x > 0 y > 2 y > 0 A triangle has its vertices at the points (1, 3), (2, 5) and (3, 4.5). Find the equations of the lines containing each side. Is the triangle a right-angled triangle? Explain how you know. What angle does the line y = 3 x + 1 make with the positive x-axis? A company delivers new cars to Doha. It has a contract to deliver at least 65 cars each day. The company owns 7 carriers that can each carry 8 cars and 5 carriers that can each carry 10 cars. The company employs 8 drivers. Each carrier can make only one journey with a full load each day. What is the maximum numbers of cars that can be delivered each day? What is the minimum number of drivers needed to fulfil the contract? 156 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.6 Algebra 2 Education Institute 2005

45 GRADE 11F: Geometry and measures 2 Applications of Pythagoras theorem UNIT 11F.7 4 hours About this unit This is the second of four units on geometry and measures for Grade 11 foundation. It focuses on the equation of a circle and other applications of Pythagoras theorem. It builds on Unit 11F.4, Geometry and measures 1. 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 10 foundation or Grade 12 foundation. 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 be able to calculate with any real numbers, including surds, using mental calculations in appropriate cases. They should be able to solve right-angled triangles using the standard trigonometric ratios, including tan θ, and Pythagoras theorem. Expectations By the end of the unit, students will use Pythagoras theorem to find the distance between two points and to set up the Cartesian equation of a circle. They will find the points of intersection of a straight line with a circle. They will approach problems systematically, knowing when it is important to enumerate all outcomes. Students who progress further will use their knowledge of the Cartesian equation of a circle to solve more complex problems. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograph (see Graphmatica (free from www8.pair.com/ksoft) computers with Internet access and graph plotting software for students graphics calculators lesson plan 11.1 Key vocabulary and technical terms Students should understand, use and spell correctly: circle, diameter, tangent, Cartesian equation, Cartesian plane, intersection, circumference, centre 157 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

46 Standards for the unit Unit 11F.7 4 hours SUPPORTING STANDARDS Grade 10F and 11F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 4 hours Applications of Pythagoras theorem 11F F.3.2 Solve right-angled triangles using the standard trigonometric ratios, including tan θ = sin θ / cos θ, and/or Pythagoras theorem. 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. 11F F.6.7 Use Pythagoras theorem to find the distance between two points in the Cartesian plane; set up the Cartesian equation of a circle of radius r, centred at the point (α, β). Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle. 10F.3.4 Calculate with any real numbers, including mental calculations in appropriate cases. 11F.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes. 158 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

47 Activities Unit 11F.7 Objectives Possible teaching activities Notes School resources 4 hours Applications of Pythagoras theorem Use Pythagoras theorem to find the distance between two points in the Cartesian plane; set up the Cartesian equation of a circle of radius r, centred at the point (α, β). Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle. Approach a problem systematically, recognising when it is important to enumerate all outcomes. Finding the equation of the unit circle Ask the class to describe precisely what is shown in the diagram. Establish that the picture shows a circle of radius 1 unit, centred at the origin. Ask students how the equation of a circle can be established. Encourage students to use a general point P with coordinates (x, y) and then use Pythagoras theorem. Continue to develop the questioning as in sample lesson plan 11.1 to look at how the angle changes and to consider whether x and y are positive or negative as the angle increases. Finding the equation of a circle with centre (α, β) Extend the simple equation of a circle to finding the equation of a circle with centre (α, β), using Pythagoras theorem to find the distance between two points. Ask students to construct a diagram to help with this, with centre (α, β) and a general point on the circle (x, y). What other information do you know? How does that help? Ask students to multiply out the general form and to see what they notice about the equation in expanded form. This column is for schools to note their own resources, e.g. textbooks, worksheets. Lesson plan 11.1 Exploring the standard equation for a circle The standard equation of a circle of radius r and centre (α, β) is (x α) 2 + (y β) 2 = r 2. Explore the effects on the circle of changing r, α and β. For example: If the value of α increases, how does the circle move? If the value of β increases, how does the circle move? What effect does decreasing the value of r have on the circle? There are some useful interactive resources for this topic on the Internet, such as that at docs/math30/conics/conics/bigcircles/localmenu.html, which allows questions such as these to be explored dynamically. Lines and circles Ask students to conjecture about all the possibilities that occur when a line and a circle are drawn on the same set of axes: they could cross, making two intersection points; they could touch so that the line is a tangent; or they could not meet at all. When you solve a problem where you are given the equation of a circle and the equation of a line, how will you know which situation applies? 159 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

48 Objectives Possible teaching activities Notes School resources Using ICT to draw circles Ask students to use a graphics calculator or a graph plotting program to produce circle patterns such as those shown. Students should print out their work and annotate it with the equations they used. What connections are there between the equations for each diagram? Practice Give students the opportunity to solve a range of problems using the equations of lines and circles. For example: Given the coordinates of two points that form the ends of a diameter of a circle, find the equation of the circle. Given the coordinates of the centre of a circle and a point on the circumference, find the equation of the circle. Why is this sufficient information to fix the circle uniquely? A circle has a radius of 10 units and passes through the points (0, 0) and (0, 12). Sketch the two possible positions of the circle and find their equations. Further problem solving Many non-routine problems depend on applications of Pythagoras theorem. These problems provide an opportunity to link geometry, algebra and trigonometry. Give students a selection, depending on their ability. Problem 1 The length of a main diagonal through the centre of a large matchbox is 6 cm. The sum of the areas of all the faces is 64 cm 2. What is the sum of the lengths of all the edges of the matchbox? Problem 1: solution Let the lengths of the three edges be a, b and c. Then the sum of the lengths of all the edges is 4(a + b + c). We are given that 2(ab + ac + bc) = 64. Using Pythagoras, a 2 + b 2 + c 2 = 36. But (a + b + c) 2 = a 2 + b 2 + c 2 + 2(ab + ac + bc). So a + b + c = ( ) = Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

49 Objectives Possible teaching activities Notes School resources Problem 2 A model boat has two sails, each on a vertical mast. Each sail is an equilateral triangle with side length 9 cm. A solution is dependent on a suitable construction. For example: Mark Z so that AZ is perpendicular to PZ. In AZP, ZP = 8 cm, AZ = 6 cm, and AZP is 90. Using Pythagoras, AP is 10 cm. The height of the taller mast is 15 cm. The distance between the bottoms of the two masts is 8 cm. What is the distance between the tips of the two sails, A and P? Problem 3 In the diagram, AD is parallel to BC, BD is perpendicular to DC, and AE is perpendicular to BD. AB is 41 cm, AD is 50 cm and BF is 9 cm. What is the area of quadrilateral DCEF? Using Pythagoras, AF = 40 cm and FD = 30 cm. Triangles ADF and BFE are similar, so FE is 12 cm. BD = BF + FD = 39 cm. Triangles BDC and BFE are similar, so DC is 52 cm. DCEF is a trapezium with height DF. Area of DCEF is 1 2 (FE + DC) DF = 960 cm 2 Problem 4 The area of an inclined face of a square-based pyramid is such that it is equal to that of a square drawn on the vertical height of the pyramid. What angle does the inclined face make with the base of the pyramid? The cosine of the required angle OPH is a/p. We are given that h 2 = ap, since the square on OH equals OAB. Using Pythagoras: In PHA, HA 2 = 2a 2 In OAH, OA 2 = h 2 + 2a 2 = ap + 2a 2 In OAP, OA 2 = p 2 + a 2 So a 2 + ap p 2 = 0 Dividing through by p 2, and solving the equation for a/p, gives a/p to be approximately OPH is thus just under Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

50 Assessment Unit 11F.7 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. 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). Two sides of a right-angled triangle are of length 21 cm and 29 cm. What are the possible lengths of the remaining side? Find the points where the line 4x 3y = 0 cuts the circle x 2 + y 2 = 100. An ellipse has the equation x 2 + y 2 = 1. The line x = y intersects the ellipse at C and D x y Which of these equations is equivalent to + = 1? Tick the correct one x + y = x + 16y = 144 Find the coordinates of C and D x + 9y = 144 x + y = 1 The side length of a cube is 10 cm. The cube is cut along a plane through three of the vertices to make a pyramid. Calculate the perimeter of the base, ABC, of the pyramid. Draw a series of diagrams with supporting text illustrating what you have learned in this unit. What are the key points? What good examples can you give? What are the main skills that you used? What things do you need to learn? Write a worksheet on the content of this unit for another student. You should make sure that you have worked solutions for all the problems you set. Swap your worksheet with that of someone else and try to solve the problems. In pairs, evaluate your worksheets. Are the questions at the right level? Were they written clearly? What were the good points of the worksheet? Where could it be improved? 162 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.7 Geometry and measures 2 Education Institute 2005

51 GRADE 11F: Number and algebra Sequences, expressions and formulae UNIT 11F.8 8 hours About this unit This is the fourth of six units on number and algebra for Grade 11 foundation. It builds on the work in Unit 11F.1, Number 1, and Units 11F.3 and 11F.6, Algebra 1 and 2. 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 10 foundation or Grade 12 foundation. 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 brackets and the order of operations when calculating. They should be able to generalise algebraic relationships to model simple patterns and sequences and to generate and rearrange formulae from a physical context. They should be able to establish and apply a formula for the sum of an arithmetic sequence, including the first n consecutive integers. Expectations By the end of the unit, students will find the sums of geometric sequences. They will simplify and combine numeric and algebraic fractions, and multiply any two monomial, binomial or trinomial expressions, collecting and simplifying similar terms. They will factorise quadratic expressions, relating the factorisation to geometric representations. They will generate formulae from physical contexts and rearrange formulae connecting two or more variables. They will use mathematics to model and predict the outcomes of real-world applications. Students who progress further will rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts. They will generate recursive sequences to model the behaviour of real-world situations. They will develop further a sense of working with symbols. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graphics calculator or computer and motion sensor balls made of different materials, measuring tapes graph paper ruler, sharp pencil, protractor and plain paper for each student Key vocabulary and technical terms Students should understand, use and spell correctly: geometric sequence, infinite geometric series, sum, common ratio formula, rearrange, substitute, expression, variable, compound interest, conversion algebraic fraction, monomial, binomial, trinomial, simplify, factorise, quadratic expression, surd, rationalise, denominator, Pascal s triangle, consecutive, Fibonacci sequence, coefficient, expand 163 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

52 Standards for the unit Unit 11F.8 8 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 11F and 12F standards 3 hours Sequences 3 hours Formulae 2 hours Algebraic manipulation 10F F F.4.9 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. Sum arithmetic sequences, including the first n consecutive integers, and give a geometric proof for the formulae for these sums. Generate formulae from a physical context; rearrange formulae connecting two or more variables. 11F F.4.7 Know the properties of geometric sequences and the conditions under which an infinite geometric series can be summed.. Generate further formulae from a physical context, and rearrange formulae connecting two or more variables; substitute an expression for a given variable into a different formula containing this variable. 12F F.4.1 Generate recursive sequences from term-to-term and position-to-term definitions to model the behaviour of real-world situations, for example population growth. Rearrange harder formulae connecting two or more variables and generate further formulae from physical contexts. 10F.4.6 Use brackets and correct order of precedence of operations when performing numerical or algebraic calculations. 11F F F F.1.2 Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions, collecting and simplifying similar terms. Factorise expressions of the form a 2 x 2 b 2 y 2, and quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions. Simplify numeric and algebraic fraction expressions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds. Use mathematics to model and predict the outcomes of realworld applications; compare and contrast two or more given models of a particular situation. 11F.4.3 Develop further a sense of working with symbols, understanding that the transformation of all such algebraic objects generalises the well-defined rules of arithmetic, and knowing 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 defined in terms of known, or given, expressions or functions 164 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

53 Activities Unit 11F.8 Objectives Possible teaching activities Notes School resources 3 hours Sequences Know the properties of geometric sequences and the conditions under which an infinite geometric series can be summed. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Bouncing a ball When a ball bounces, it rebounds each time to a fixed proportion of its previous height. Hence the heights of the bounces form a geometric sequence with common ratio r, which is known as the rebound rate. Ask students to work in small groups to collect data by bouncing a ball and measuring the height of each successive bounce. This can be done either using a ball dropped in front of a scale stuck on the wall, or using a motion sensor connected to a computer or a graphics calculator. After they have collected the data, ask students to work out the common ratio r for the sequence formed. The task can be extended to answer questions such as: What effect does the initial height have on the rebound height? Try balls of made of different materials. What effect does that have on the rebound height? What other factors affect the way the ball rebounds? Solve word problems based on other practical situations giving rise to a geometric sequence, for example: the decaying of a pendulum swing; diluting a solution by removing half each time and examining the concentration. Snowflakes Draw an equilateral triangle with sides 9 cm long. Divide each side into three and construct another equilateral triangle on the middle third of each side. Repeat the process to form the third snowflake design and again for the fourth, drawing equilateral triangles on the middle third of each side of the previous design. Calculate the length of the perimeter for each of the designs. What happens to the length of the perimeter as the number of steps increases? What happens to the area of the snowflake? This column is for schools to note their own resources, e.g. textbooks, worksheets. 165 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

54 Objectives Possible teaching activities Notes School resources Sum of a geometric sequence Work through with students the sum to n terms of a geometric sequence. Assume the sequence is S = a + ar + ar ar n 1 Multiplying by r gives Sr = ar + ar ar n Subtracting gives S(1 r) = a(1 r n ) So S = a(1 r n )/(1 r) Discuss conditions under which a geometric sequence can be summed to infinity. Investigating compound interest as an example of a geometric sequence Write a general expression for saving QR x at an annual rate of y% compound interest for n years. Graph an example of saving an initial sum at a fixed rate and describe how the money increases. Compare this with the same rate as simple interest. The first three terms of an arithmetic sequence a, a + d, a + 2d are the same as the first three terms of a geometric sequence, a, ar and ar 2. For what values of r and d is this possible? 3 hours Formulae Generate further formulae from a physical context, and rearrange formulae connecting two or more variables; substitute an expression for a given variable into a different formula containing this variable. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Formulae from physical contexts Generate and use formulae from physical contexts such as: The formula for the volume V of a square-based pyramid is V = 1 3 b 2 h, where b is the base length and h is the perpendicular height. A square-based pyramid has base length 5 cm and perpendicular height 6 cm. What is its volume? A different square-based pyramid has base length 4 cm. Its volume is 48 cm 3. What is its perpendicular height? The volume of another square-based pyramid is 25 cm 3. Its perpendicular height is 12 cm. What is its base length? The second diagram shows a triangular-based pyramid. The base is an isosceles rightangled triangle. The perpendicular height is m. Write a formula, in terms of m, for the volume V of the pyramid. The diagram shows a square, ABCD, of side length (p + q). Inside it is a square, EFGH, of side length r. Write a simplified expression for: the area of square EFGH (in terms of r); the area of triangle DEH (in terms of p and q); the area of square ABCD (in terms of p and q). Square ABCD is made up from four triangles and square EFGH. The four triangles are all congruent to triangle DEH. Use this information to write an expression in terms of p, q and r for the area of square ABCD. Use the two different expressions for the area of square ABCD to express r 2 in terms of p 2 and q Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

55 Objectives Possible teaching activities Notes School resources A solid cuboid, a cm by b cm by c cm, is made out of 1 cm cubes. a, b and c are all greater than or equal to 2. The outside of the cuboid is covered completely in green paint. The eight corner cubes have three faces painted green. Write an expression for the number of 1 cm cubes that have exactly: two faces painted green; one face painted green; no faces painted green. If a = b = c, the cuboid is a cube of side a cm. The cube is also covered in green paint. The table shows how many 1 cm cubes have 0, 1, 2 or 3 faces covered. No. of faces covered No. of 1 cm cubes (a 2) 1 6(a 2) 2 0 (a 2) 3 The total number of 1 cm cubes in the cubes is a 3. You can tell from the table that: (a 2) + 6(a 2) 2 + (a 2) 3 = a 3 Use an algebraic method to show that this is true. Formulae from science Rearrange and use formulae from science, such as: The time for a pendulum swing is given by T 2 = 4π 2 l/g. Write the formula in terms of l. h = 1 2 gt 2. Rewrite this to give t in terms of g and h. E k = 1 2 mv 2. What is v in terms of the other variables? F g = Gm 1 m 2 /r 2 is a commonly used equation in physics. What other rearrangements can you make that are equivalent? There are five constant acceleration formulae commonly used in physics. These are: v 2 u 2 = 2as d = u + at s = ut at 2 s = vt 1 2 at 2 s = 1 2 (u + v)t What connections can you find between the equations? Can you use any two of them to derive a third? Practice Provide some practice in using and applying the above formulae. 167 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

56 Objectives Possible teaching activities Notes School resources The formula that connects the volume V of an air bubble at the water surface, its initial volume U, and the distance d it has risen is: d V = 10.3 U where V and U are in cm 3 and d is in metres. At 50 metres below the water surface a bubble has a volume of cm 3. What is the volume of the bubble at the water surface? The volume of a bubble at the water surface is 4 times its initial volume. By how many metres has the bubble risen? At the water surface a bubble has a volume of cm 3. What was the initial volume of this bubble when it was 20 m below the water surface? A spherical bubble, 35 metres below the water surface, had a radius of cm. Calculate the volume of this bubble using the formula: volume = 4 3 πr 3 What is the radius of this bubble at the water surface? The mean maximum daily temperature is given below for several countries. Students should give their answers to this problem in standard form. Country Doha, Qatar Bahrain Muscat, Oman Maddinah, Saudi Arabia Temperature 41.5 C F 40.3 C F To convert from Centigrade to Fahrenheit, use the formula F = 9 5 C + 32, where C is the temperature in degrees Centigrade and F the temperature in degrees Fahrenheit. Use the formula to convert the temperatures above so you can compare them. Rearrange the formula to convert the temperatures from Fahrenheit to Centigrade. Devise an approximate formula for the conversion. Does this formula ever give the exact answer? For what range of temperatures does your approximate formula give an answer within 5 F? Do you think that your approximate formula is accurate enough for the Middle East? 168 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

57 Objectives Possible teaching activities Notes School resources 2 hours Algebraic manipulation Combine numeric or algebraic fractions, and multiply combinations of monomial, binomial and trinomial expressions, collecting and simplifying similar terms. Factorise expressions of the form a 2 x 2 b 2 y 2, and quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions. Simplify numeric and algebraic fraction expressions by cancelling common factors; rationalise a denominator of a fraction when the denominator contains simple combinations of surds. Algebraic manipulation Consolidate: multiplying out expressions; factorising quadratic expressions; simplifying algebraic expressions. Wherever possible, use diagrams to support the work. For example, ask students to use the diagram on the right to explain why (x 2)(x 1) = x 2 3x +2. Students should write very clear explanations and then discuss them in pairs in order to produce better explanations. Construct some quadratic expressions from two linear factors in a and b and draw geometric representations for them. Expression snap Prepare a set of cards with expressions on them, some of which should be equivalent. Give a set of cards to each pair of students. The cards should be shuffled and then all shared between the two players. Each player keeps their cards face down in a pile. Player 1 turns over their top card and uses it to create a new pile. Player 2 then turns over their top card and places it on top of the new pile created by player 1. The players take it in turns to add a card to the pile until a card is placed on top of another card bearing the equivalent expression. The first player to notice the match shouts Snap! and wins the whole pile of cards. The winner is the player who finishes with the whole pack of cards. Pascal s triangle Pascal s triangle was originally used by the ancient Chinese, but Blaise Pascal was the first person to realise the uses of the many patterns it contains. Show students how to construct Pascal s triangle and ask them to draw a copy for themselves on a large piece of paper. Ask them to look for the patterns it contains and to link those patterns to the areas of mathematics they have studied. For example, Where do the consecutive integers appear? Where can you find the triangular numbers? Can you find the Fibonacci sequence? Write out the expansions for (x + 1) 0, (x + 1) 1, (x + 1) 2, (x + 1) 3, (x + 1) 4, What link can you find between these expansions and Pascal s triangle? This could be done either to match factorised and expanded versions of quadratics or with expressions involving surds to match those with rationalised denominators and those with irrational denominators. There are lots of websites containing interactive versions of Pascal s triangle which help students to find the patterns, e.g. mathforum.org/workshops/usi/pascal/ mo.pascal.html 169 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

58 Assessment Unit 11F.8 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. 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. 1 kilogram of rice contains approximately grains of rice. Estimate the weight of all the rice on the chessboard. The sum of the infinite geometric series is A. 5 8 B. 2 3 C. 3 5 D. 3 2 Melons cost QR 1.5 each and apples cost QR 3.75 per kilogram. A man buys apples and melons at the supermarket. Write a formula to describe the total cost of his purchase. Investigate how many melons and how many kilograms of apples he 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. Find R in terms of R 1 and R 2 when 1/R = 1/R 1 + 1/R 2. Use Pascal s triangle to read off the coefficients of the powers of x in the expansion of (1 + x) n for different values of the positive integer n. Check the results for n = 3 by expanding (1 + x) 3. Simplify (2x 3)(x 2 + x 10). Without using a calculator, find the exact value of Explain why (a + b) 2 a 2 + b 2. Draw a diagram to represent the identity (a + b) 2 = a 2 + 2ab + b 2. A curve touches a circle of radius a, centre (0, a). The equation of the curve is: 3 8a y = 2 2 x + 4a When a = 5, what is the maximum value of y? When a = 2 and y = 1.5, what are the values of x? When y = 2 5 a, what, in terms of a, are the values of x? Rationalise the expression 1/( 2 + 3). 170 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.8 Number and algebra Education Institute 2005

59 GRADE 11F: Geometry and measures 3 Circular functions and trigonometry UNIT 11F.9 5 hours About this unit This is the third of four units on geometry and measures for Grade 11 foundation. 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 in this unit should help you to plan the content, pace and level of difficulty of lessons. Adapt the ideas to meet your students needs. For consolidation or extension activities, look at the units for Grade 10 foundation or Grade 12 foundation. 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 Pythagoras theorem and the standard trigonometric ratios. Expectations By the end of the unit, students will use Pythagoras theorem to show that sin 2 θ + cos 2 θ = 1 for any angle θ. They will plot the graphs of circular functions and solve simple problems modelled by these functions. They will develop chains of logical reasoning, using correct notation and terms, and will explain their reasoning, both orally and in writing. Students who progress further will use their knowledge of circular functions to solve more complex problems. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograph (see Graphmatica (free from www8.pair.com/ksoft) computers with graph plotting software for students graphics calculators for students graph paper trigonometric matching cards lesson plan 11.1 Key vocabulary and technical terms Students should understand, use and spell correctly: sine, cosine, tangent, trigonometric functions, periodic motion, unit circle, variation Pythagoras theorem, trigonometric equation, inverse trigonometric function, arcsine, arccosine, sin 2 θ, cos 2 θ 171 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

60 Standards for the unit Unit 11F.9 5 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 2 hours Circular functions 3 hours Trigonometry 10F F.6.5 Use Pythagoras theorem to find the distance between two points and to solve right-angled triangles; set up the Cartesian equation of a circle of radius r, centred at the origin of an xy-coordinate system. 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; use these ratios to find the angles of a right-angled triangle given two sides, or to find the remaining sides given one side and one angle. 11F F.6.10 Use the unit circle x 2 + y 2 = 1 to plot graphs of the circular functions sin θ and cos θ for any angle θ, where 0 θ 360 ; know that any point on this circle has coordinates (cos θ, sin θ ), where θ is the angle the radius to the point makes with the positive x-axis. Use a calculator to find sine and cosine values of a given angle and to find the angle corresponding to a given value of the sine or cosine of that angle, and know that these are inverse functions defined on a restricted domain. 11F.6.11 Use Pythagoras theorem to show that sin 2 θ + cos 2 θ = 1 for any angle θ. 11F.6.12 Solve simple problems modelled by circular functions. 11F.1.6 Develop chains of logical reasoning, using correct mathematical notation and terms. 11F.1.7 Explain their reasoning, both orally and in writing. 172 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

61 Activities Unit 11F.9 Objectives Possible teaching activities Notes School resources 2 hours Circular functions Use the unit circle x 2 + y 2 = 1 to plot graphs of the circular functions sin θ and cos θ for any angle θ, where 0 θ 360 ; know that any point on this circle has coordinates (cos θ, sin θ ), where θ is the angle the radius to the point makes with the positive x-axis. Solve simple problems modelled by circular functions. Explain their reasoning, both orally and in writing. Tides Tide tables can be produced by modelling the variation in the height of the water above a fixed level using sine and cosine functions. Here is a simple example. The tidal heights above mean low water for the Marshall Islands over a 24-hour period are as follows: Time Height (m) Time Height (m) Time Height (m) Time Height (m) y = 0.6 sin (0.4x) superimposed on raw data This column is for schools to note their own resources, e.g. textbooks, worksheets Why are trigonometric functions useful for modelling tidal heights? Plot the graph for a set of tide data and see if you can fit a sine function to it. A computer or graphics calculator might be useful here. Can you explain how the equation is formed and the significance of the different parts? Why does the height of the water go up and down in tidal areas? Ferris wheel The first Ferris wheel was designed by George Ferris in It had a radius of approximately 40 metres and took 20 minutes to complete a full turn. When will a person be three quarters of the way up? Suppose Mr Ferris designed another wheel that rotated twice as fast. When will a person be three quarters of the way up now? NOTE: This activity emphasises the periodic nature of such movement, when solutions continue to appear at regular intervals as the motion continues. 173 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

62 Objectives Possible teaching activities Notes School resources Investigating how the coordinates of a point change as it moves round a circle Ask students to mark the position of point P every 15 as it moves round the circle. Lesson plan 11.1 Use the diagram to mark the position of point P on a set of axes by plotting y against the angle θ. Question students as suggested in lesson plan What is the equation of this curve for y in terms of θ? How should the task be adapted to show the variation in the x-coordinate of P as θ varies? What is the equation of this variation? What is the relationship between the two graphs? Ask students to use an interactive graphing software package or a graphics calculator to investigate the following. Lesson plan 11.1 Investigate the graph of the function y = tan x, where tan x = sin x / cos x. Explore the effect of working on a circle of radius A and the effect of setting the zero for the angle θ at a different position. Explore the effect of doubling or halving the argument of the function, i.e. the definition of the angle being measured. 3 hours Trigonometry Use a calculator to find sine and cosine values of a given angle and to find the angle corresponding to a given value of the sine or cosine of that angle, and know that these are inverse functions defined on a restricted domain. Use Pythagoras theorem to show that sin 2 θ + cos 2 θ = 1 for any angle θ. Develop chains of logical reasoning, using correct mathematical notation and terms. Investigating sine, cosine and tangent On the diagram write expressions for the lengths of the unknown sides. What form does Pythagoras theorem take using this diagram? Now challenge students to find other values of θ where sin θ = 0.5. How many can you find? Can you connect these with previous graphs you have drawn? What properties of the graph explain this pattern? Can you write a rule to generalise what these angles are? Repeat this activity for cosine and tangent and for different values. Practice: matching cards Prepare a set of cards where each card has at least one matching partner using some of the key points from this unit and preceding trigonometry and circular function units. Some suitable cards are shown on the right. Each pair of students has a set of cards, The whole set of cards is shuffled and spread out on a table face down. Students take turns to turn over any two cards looking for a matching pair. If the pair does not match the cards are replaced face down in the same position. If the cards do match, the student explains to the other player why they match and the pair is put on one side. The winner is the player who matches the most cards. Double points are awarded for making triples or larger sets of cards. There are a number of interactive resources on the Internet that would help with this activity, for example: sin 2 θ + cos 2 θ 1 sin θ = 0.5 θ = 30 θ = 150 θ = Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

63 Objectives Possible teaching activities Notes School resources Use a calculator or computer and the inverse trigonometric functions arcsine and arccosine. What values of sine are impossible? What values of cosine are impossible? What are the restricted domains for these functions? Practice Give students some triangles to solve using the arcsine and arccosine functions. Investigating overlapping squares Use two identical squares of paper. Place one exactly on top of the other. As the top square is rotated about its centre, the area of overlap between the squares decreases, and the corners of the lower square stick out or protrude. Ask: What is the largest fraction of the lower square that you will see? There are various approaches. One is through measuring and calculating areas of triangles. A more thorough investigation uses algebra, Pythagoras theorem and surds. Ask: What angle of rotation gives the largest area of the lower square visible, and what sort of proof is needed? Get students to measure the sides of the four protruding triangles, and calculate their total area, expressing this as a fraction or percentage of the lower square. Then ask: What would happen to this fraction if the original squares were a different size? What sort of proof is needed? An alternative approach is to assume from symmetry that 45 will give the largest fraction, then PQ = x 2. The side of the square is 2x + x 2 = 1, so x = 1/(2 + 2), and the fraction of the total area that is visible is 2/(2 + 2) 2. A useful resource here is You could introduce this investigation using dynamic geometry software (DGS), such as Geometer s sketchpad. For able students, extend the problem to the equivalent problem with equilateral triangles or other regular polygons. What would happen to the visible fraction as we increase the number of sides of the original polygon? What sort of arguments can be offered? If 45 is not assumed, then trigonometry is needed to describe the link between the protruding edges of the square The final fraction, for any angle of rotation t is: 2sin t cos t / (1 + sin t + cos t) 2 which can be plotted to show the maximum value at 45. Substitution into the expression gives the same result as above. 175 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

64 Assessment Unit 11F.9 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Explain why sin (180 θ ) = sin θ. Give the exact value of cos 225. What other angle has the same cosine value? sin 2 θ + cos 2 θ = 1 Verify this result for the angles 30, 45 and 60. What happens when θ = 90? Find the angle whose sine is What is the cosine of this angle? For this angle, verify that sin 2 θ + cos 2 θ = 1. Produce a revision booklet for this topic for another student. Make sure all the key facts are included and illustrated with diagrams and examples. Swap guides with another student and compare your two guides. Did you choose the same key facts? What examples and diagrams did you each choose to include? 176 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.9 Geometry and measures 3 Education Institute 2005

65 GRADE 11F: Algebra 3 Quadratic equations UNIT 11F.10 9 hours About this unit This is the fifth of six units on number and algebra for Grade 11 foundation. It is the third in a series of four units on algebra and builds on Units 11F.3 and 11F.6, Algebra 1 and 2. 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 10 foundation or Grade 12 foundation. 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 plot straight line graphs of the form y = mx + c, and to relate the gradient and intercepts on the axes to m and c. They should be able to identify the equation of a straight line from its graph alone, or from the coordinates of two points on the line, or from the coordinates of one point on the line and the gradient of the line. Expectations By the end of the unit, students will, through their continued study of quadratic functions and their graphs, and the solution of associated equations, appreciate a range of numerical and algebraic applications in the real world. They will model situations with quadratic functions and find exact and approximate solutions of quadratic equations. Students will use mathematics to model and predict the outcomes of real-world applications. Students who progress further will apply combinations of transformations to the graph of the function y = f(x). Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograph (see Graphmatica (free from www8.pair.com/ksoft) graphics calculators for students graph paper lesson plan 11.2 Key vocabulary and technical terms Students should understand, use and spell correctly: quadratic function, completing the square, quadratic formula, roots, range axis of symmetry, turning point, coefficient, minimum, maximum, parabola, reflection approximate solution, intersection point, equivalent 177 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

66 Standards for the unit Unit 11F.10 9 hours SUPPORTING STANDARDS Grade 10F and 11F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 9 hours Quadratic equations 11F.4.5 Factorise expressions of the form a 2 x 2 b 2 y 2, and quadratic expressions; conceptualise geometric representations for these factorisations and other similar quadratic expressions. 11F.5.10 Model a range of situations with appropriate quadratic functions. 10F.5.14 Recognise a simple second-order polynomial in one variable, y = ax 2 + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas), and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point. Recognise a second-order polynomial in one variable, y = ax 2 + bx + c, as a quadratic function; plot graphs of such functions (recognising that these are all parabolas), and pick out the intercepts with the coordinate axes, the axis of symmetry and the coordinates of the maximum or minimum point; understand when such functions are increasing, when they are decreasing and when they are stationary. 11F.5.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula. 11F F F F.1.2 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 ax 2 + bx + c. Find approximate solutions of the quadratic equation ax 2 + bx + c = 0 by reading from the graph of y = ax 2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. 12F.5.7 Understand the transformations of the function y = f(x) given by: y = f(x) + a, representing a translation by a in the positive y-direction; y = f(x a), representing a translation by a in the positive x-direction; y = a f(x), representing a stretch with scale factor a parallel to the y-axis; y = f(ax), representing a stretch with scale factor 1/a parallel to the x-axis; use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function. 178 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

67 Activities Unit 11F.10 Objectives Possible teaching activities Notes School resources 9 hours Quadratic functions Model a range of situations with appropriate quadratic functions. Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula. 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 ax 2 + bx + c. Find approximate solutions of the quadratic equation ax 2 + bx + c = 0 by reading from the graph of y = ax 2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Modelling with quadratic equations Invite students to work with a number of contexts where real-life data can be modelled using a quadratic function. These might include: rolling a ball diagonally across an inclined plane; the motion of a body moving with constant acceleration to see how the distance moved depends on the time taken. Ask them to investigate sets of data to see if a quadratic model fits better than a linear one. Examples might include: quantity of waste put in landfill sites over time; fuel efficiency of cars over decades; trade figures over time. Problem solving A courtyard garden is 12 m by 10 m. The fountain feature in the middle is surrounded by paving on both long sides and one short side such that the area of the paving is half that taken up by the water feature. What is the width of the paving? A rectangle has a perimeter of 54 cm and an area of 180 cm 2. Using a quadratic equation, find the lengths of the sides. Write an expression for the surface area of a cylinder of radius r and height 12 cm. If the surface area of the cylinder is 534 cm 3, what is the radius? Graphs, factors and solutions Ask students to factorise the expression x 2 4x + 3. What are the solutions to the quadratic equation x 2 4x + 3 = 0? Draw the graph of the quadratic function y = x 2 4x + 3. Where on the graph are the solutions of the quadratic equation x 2 4x + 3 = 0? Add the two linear functions y = x 1 and y = x 3 to the graph. Why did we choose these two linear functions? What do you notice about where these linear functions cross the quadratic function? Does this happen with other quadratic expressions? If so, can you explain why? Encourage students to make connections between solutions arrived at algebraically and their graphical representations. This column is for schools to note their own resources, e.g. textbooks, worksheets. Lesson plan 11.2 It is also useful to plot y = (x 1)(x 3) to demonstrate that the two forms of the quadratic are equivalent. 179 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

68 Objectives Possible teaching activities Notes School resources Exploring coefficients Challenge students to use a graphics calculator to find out how the coefficients a, b and c affect the graph of y = ax 2 + bx + c. Remind them of the benefit of only changing one variable at a time. Ask students to explore questions such as: How can you move the function y = x 2 up the vertical axis? How can you move the function y = x 2 down the vertical axis? What effect does a have on the shape of the graph? What happens if a is negative? Can you find any rules for how the graph of the function behaves as b changes? Ask students to make sketches as they carry out their investigation and to produce a written report at the end. Solving problems in more than one way Ask students to find as many ways as they can to solve the problem of finding a number where the difference between that number and its square is 1. This situation can be written algebraically as the quadratic equation x 2 x = 1. What ways can you find to solve this? In how many different ways can you represent the solution graphically? Ask them also to discuss: Which ways do you prefer? What are the advantages of the different methods? This investigation can be carried out using a graphics calculator or a computer with graph plotting software. Alternatively, useful applets are on websites such as members.shaw.ca/ron.blond/qfa.csf.applet/index.html. An alternative approach is to consider how to shift the graph to the left and right by using the quadratic function in factorised form, e.g. using y = (x 3) 2 and y = (x + 3) 2 to shift right three units and left three units. This can lead to a discussion about why this must be true, by considering the roots of the equation. A graphical solution can be found and illustrated by: x 2 x = 1 x 2 x 1 = 0 x 2 = x + 1 Practice: matching graphs Give students some graphs of quadratic functions and a list of possible functions and ask them to match the function to the graph to make a pair. This could be done using printed cards, but also by using printouts of graphics calculator screens reproduced on acetate or by using web-based resources such as those at and Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

69 Objectives Possible teaching activities Notes School resources Completing the square Ask students to look for general rules to work out how to write quadratic expressions as perfect squares or in the form (x + a) 2 + b in order to complete the square. For example: What do you notice about these equivalent forms? x 2 20x (x 10) 2 x x + 25 (x + 5) 2 Or these? x 2 49 (x 7)(x + 7) x 2 64 (x 8)(x + 8) Or these? x 2 6x + 2 (x 3) 2 7 x 2 + 8x + 12 (x + 4) 2 4 Can you expand this method to quadratic expressions with an odd number for the coefficient of x and for coefficients of x 2 other than 1? Students should then consider the graphs of these functions. How do the functions in completed square form relate to their key features (axis of symmetry, roots and minimum value)? How can writing quadratic equations in this form help you to solve them? Ask students to take the method of completing the square used in the above example and use it on the general quadratic equation ax 2 + bx + c = 0. What do you get? What can you use it for? How can you tell when there are no solutions to an equation? How can you tell when there is only one solution to an equation? (the root is repeated) What is the graphical interpretation of each of these situations? Practice Give a short exercise in completing the square. Ask students to pick a quadratic function and to plot it on graph paper. Now ask them to exchange graphs with another student. Each student writes five different equations that can be solved using the graph they have been given. Next they exchange the graphs back, solve the equations and then discuss their solutions in pairs. For example, the function plotted is y = x 2 6x + 2. Three possible equations to solve are: x 2 6x + 2 = 0 x 2 6x = 7 x 2 6x = Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

70 Assessment Unit 11F.10 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. A fountain at ground level sprays out jets of water. Each jet is a parabola. The jet that sprays the farthest has equation y = x 2 + 8x 15. Factorise this expression. Hence find: a. where the fountain jet is positioned in this xy-coordinate system, and b. how far from the fountain jet the water hits the ground. Calculate the greatest height that the water reaches. Huda throws a ball to Mariam who is standing 20 m away. The ball is thrown and caught at a height of 2.0 m above the ground. The ball follows the curve with equation y = 6 + c(10 x) 2, where c is a constant. Calculate the value of c by substituting x = 0, y = 2 into the equation. Curve A is the reflection in the x-axis of y = x 2. What is the equation of curve A? An n-sided polygon has 1 2 n(n 3) diagonals. How many diagonals has an octagon? A polygon has 104 diagonals. How many sides does it have? y = (x 3) is a quadratic function of x. What is the minimum value of this function and for what value of x does it occur? What is the maximum range of the function? Give the equation of the axis of symmetry of the function. Write an alternative form for the equation defining the function. Sketch the graph of this function. The graph on the right shows the curve y = x 2 + 4x. a. Solve the equation x 2 + 4x 2 = 0 using the graph. Give your answers to two decimal places. b. The equation x 2 + 4x + 5 = 0 cannot be solved using the graph. Why not? 182 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.10 Algebra 3 Education Institute 2005

71 GRADE 11F: Statistics 2 Representing and interpreting data UNIT 11F.11 7 hours About this unit This is the second of two units on statistics for Grade 11. It builds on Unit 11F.5, Statistics 1. The main emphasis in this unit is on students carrying out their own projects. The unit suggests a number of projects and some possible lines of enquiry. 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 10 foundation or Grade 12 foundation. 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 formulate a problem, collect and analyse data and draw conclusions in a range of simple situations. They should be able to calculate the mean and median of sets of data. Expectations By the end of the unit, students will continue to calculate and use measures of central tendency. They will analyse results to draw conclusions, and will use a range of graphs, charts and tables to present their findings. Students will be able to explain their reasoning, both orally and in writing. They will synthesise, present, interpret and criticise mathematical information, and recognise when to use ICT and do so efficiently. Students who progress further will calculate measures of spread, including the variance and standard deviation. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector spreadsheet software such as Microsoft Excel statistics software such as Autograph (see computers with Internet access, spreadsheet and statistics software for students graphics calculators for students Key vocabulary and technical terms Students should understand, use and spell correctly: histogram, frequency, (cumulative) frequency distribution, frequency density, relative frequency, relative frequency distribution, range, percentile, interquartile range, semi-interquartile range, average, mean, mode, modal class, modal frequency, variable, scatter graph, correlation stem-and-leaf diagram, stem plot, box-and-whisker plot, box plot random sample, variation, population, sample, representative sample, questionnaire, survey, experiment, primary data, secondary data, hypothesis, bias 183 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

72 Standards for the unit Unit 11F.11 7 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 7 hours Representing and interpreting data 10F F.8.5 Calculate measures of central tendency such as the arithmetic mean and the median. Make simple inferences and draw conclusions from the formulation of a problem to the analysis of data in a range of simple situations. 11F F.8.6 Calculate and use measures of central tendency such as the arithmetic mean and the median. Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings. 12F.10.5 Calculate measures of spread, including the variance and standard deviation. 11F.9.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT applications to present statistical tables and graphs. 11F.1.7 Explain their reasoning, both orally and in writing. 11F.1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms. 11F.1.14 Recognise when to use ICT and when not to, and use it efficiently. 184 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

73 Activities Unit 11F.11 Objectives Possible teaching activities Notes School resources 7 hours Representing and interpreting data Calculate and use measures of central tendency such as the arithmetic mean and the median. Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; use a range of statistics and graphs, charts and tables to present and justify findings. Use a calculator with statistical functions to aid the analysis of large data sets, and ICT applications to present statistical tables and graphs. Explain their reasoning, both orally and in writing. Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms. Recognise when to use ICT and when not to, and use it efficiently. Statistical investigation The main emphasis in this unit is on students carrying out their own projects. The unit suggests a number of projects and some possible lines of enquiry. Ask students to design their own statistical project around their own interests. Students should be allowed to choose, perhaps from a shortlist, a project of interest to themselves, using either primary or secondary data. They should be encouraged to use appropriately and develop more confidence with those statistical enquiry techniques they have already met and to use ICT where it will be of benefit. The project could be in the field of: sport: for example, analysis of results between competitors or over time; geography: for example, making comparisons of climate or population data between countries; medicine: for example, the spread or eradication of disease or illness; economics: for example, either comparing countries or examining change over time; the environment: for example, carbon dioxide or carbon emissions over time. Some examples of starting points for projects are given below. All the suggested projects are suitable for students to work on in groups. The Olympics The Olympics is the premier sporting event in the world. Ask students to think of questions about the Olympics that they might like to find out the answers to. What hypotheses can they form? For example: Do winning margins in athletics get smaller and smaller? How good will a sports person need to be to get a gold medal in 2008? Is it now easier to get a perfect score in gymnastics than it used to be? Is China becoming a more dominant force at the Olympics? If so, which nations are doing less well than they used to? Now ask students to make a plan and decide what data they need to collect. They should refine their hypothesis so that it is clear and so that they know what they are measuring and what information they will need. At the planning stage, they also need to think about how they will analyse the data. A lot of data is available about the Olympics in books and on the Internet. For example: gbrathletics.com/olympic exploringdata.cqu.edu.au/datasets.htm Ask students to analyse their data and to report their findings. They should write about their findings and clearly link their work back to their original hypothesis. Useful websites include: The CIA World Factbook Census at School censusatschool.ntu.ac.uk Exploring data exploringdata.cqu.edu.au The graph shows the winning distance in the men s long jump plotted against the year of the games. It shows that in some years the winning distance was shorter than that of the previous games, but that the general trend is upwards, so in order to win the gold medal an athlete must expect to jump approximately 2.8 cm further than the previous gold medallist. This column is for schools to note their own resources, e.g. textbooks, worksheets. 185 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

74 Objectives Possible teaching activities Notes School resources Health and disease The field of heath and disease is a rich area for statistical investigation by students. Two such possibilities are the spread of AIDS and HIV infection and the analysis of data on smoking and lung cancer. What follows is a sample investigation using data from exploringdata.cqu.edu.au/datasets.htm. The data in the spreadsheet on the web page gives a smoking index. This is 100 if the people in an occupation are exactly average in their smoking. It is below 100 if they smoke less than average, and above 100 if they smoke more than average. The data also gives a cancer index. This is greater or less than 100 in the same way, depending on whether there are more or fewer deaths from lung cancer than would be expected. Ask students to use a spreadsheet to analyse data on the mortality and smoking indices. They need first to take a careful look at the data and interpret the highest and lowest indices for each variable. Can you see any relationships between occupational group and one of the indices? What factors might explain this? Students should then be encouraged to look at the data as linked pairs. Can you find any relationship between the smoking and mortality indices? The scatter graph shows mortality index plotted against smoking index. It shows that there is a weak positive correlation between smoking and lung cancer, so that those people who smoke are more likely to die from lung cancer. Global warming Many people believe that the Earth s temperature is getting warmer, leading to changes in climate that could have a major impact on our environment and the way we live. Ask students to list some of the current environmental issues in Qatar. These might include: increased dependence on large-scale desalination facilities for fresh water; air pollution; marine pollution. Ask students to use the Internet to find some data about climate change. Useful sites include: yosemite.epa.gov/oar/globalwarming.nsf/content/resourcecenterdata.html exploringdata.cqu.edu.au/datasets.htm Ask students to use the information they find to see what changes, if any, there have been. They will need to decide what information to use, over what period of time and for what part of the world. Encourage students to comment critically on what they find out. 186 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

75 Objectives Possible teaching activities Notes School resources Crude oil production The Middle East Economic Survey publishes figures for PREC crude oil production for 1999 through to 2004 ( Ask students to use a spreadsheet or statistics package to find the mean monthly output per country in each year and to set themselves some questions to answer using the data. For example: How has the oil output from Qatar changed over the last five years? Are there any unusual changes in the patterns of oil production for any countries? Can you explain these? Has the ranking of countries in terms of their oil production changed over time? Ask students to produce evidence and explanations backing up the answers to their questions. The graph shows oil production in Qatar over the first 11 months of There was an increase in production over the period January to August with production then leveling off. Overall growth is just over 14% but it looks greater on the graph because the vertical axis scale does not start at zero. Body measurements There must be some basic relationships between our body measurements, otherwise everyone would look completely different. Ask students what connections they can think of. For example: Are all of us six feet high? That is, is our height six times the length of our feet? Do those who are taller have a longer stride? Is height related to the length of your middle finger? How do clothes manufacturers design clothes for the average person? Ask students to work in groups to plan their study, paying particular attention to forming a clear hypothesis and using an appropriate sample. Experiments Another good source of data for statistical investigation is data collection by an experiment. Two possibilities are to examine students estimation skills or their ability to memorise something. Ask students to brainstorm these or other areas for investigation. What could you ask other students to estimate? What could you ask other students to memorise? What factors might affect a person s ability to estimate/memorise? What hypotheses might you test? What variation do you think there is in a person s ability to estimate or memorise? Students should then design an experiment to test their hypothesis, paying particular attention to the selection of a sample that will reflect their chosen population. Suitable experiments might include estimating a period of time or estimating an angle or length. Students could design a list of nonsense words and a list of meaningful words for a memory test. Factors to consider might include age and gender. 187 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

76 Assessment Unit 11F.11 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Investigate life expectancy in a range of countries including Qatar, Iran, Turkey, India, Brazil, China, Russia, Italy, the United Kingdom and the United States of America. Develop criteria for assessing a statistical investigation from this unit. What are the most important things that should have been considered? What extra things improve a project? What things let a project down? Use the criteria to grade your own work, then write down three things you did well and three things that could be improved. Grade another student s project and give constructive criticism. Prepare a presentation for the whole of the class on your project and present the findings to the group. Alternatively, produce one side of A4 paper summarising the key points of the project, or a poster for display. Make a glossary of terms and techniques in statistics. Each entry should be illustrated with an example. 188 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.11 Statistics 2 Education Institute 2005

77 GRADE 11F: Geometry 2 Circle theorems UNIT 11F.12 4 hours About this unit This is the second of two units on geometry for Grade 11 foundation. It builds on the work in Unit 11F.2, Geometry 1. 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 10 foundation or Grade 12 foundation. 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 angles at a point, angles on a straight line, and alternate and corresponding angles to establish the congruency of two triangles. They should be able to use a dynamic geometry system to conjecture results and to explore geometric proof. Expectations By the end of the unit, students will prove and use standard circle theorems. They will generate mathematical proofs and identify exceptional cases. They will develop and explain chains of logical reasoning, using correct mathematical notation and terms. Students who progress further will use their knowledge of circle theorems to solve more complex problems. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector dynamic geometry system (DGS) such as: Cabri Geometrie ( Geometer s sketchpad ( computers with Internet access and dynamic geometry system (DGS) software for students sharp pencil, straight edge (ruler), compasses and protractor for each student Key vocabulary and technical terms Students should understand, use and spell correctly: parallel, perpendicular, centre, chord, bisect, tangent, subtend, arc, semicircle, segment, intersect, cyclic quadrilateral, supplementary radius, diameter, circumference, arc length, sector, inscribed circle, circumcircle, convex 189 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

78 Standards for the unit Unit 11F.12 4 hours SUPPORTING STANDARDS Grade 10F and 11F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 4 hours Circle theorems 10F.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. 11F.6.14 Prove and use the theorems: The perpendicular from the centre of a circle to a chord bisects the chord. The two tangents from an external point to a circle are of equal length. 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 a 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. 11F.6.1 Use dynamic geometry systems to conjecture results and to explore geometric proof. 11F1.8 Generate mathematical proofs, and identify exceptional cases. 11F.1.6 Develop chains of logical reasoning, using correct mathematical notation and terms. 190 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

79 Activities Unit 11F.12 Objectives Possible teaching activities Notes School resources 4 hours Circle theorems Prove and use the theorems: The perpendicular from the centre of a circle to a chord bisects the chord. The two tangents from an external point to a circle are of equal length. 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 a 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. Generate mathematical proofs, and identify exceptional cases. [continued] Using DGS to explore circle theorems Use a dynamic geometry system (DGS) to explore constructions relating to circle theorems. Students could either be given prepared files or asked to make some of their own. For example, to explore the relationship between the angle subtended by an arc at the centre of a circle and the angle subtended by the arc at the circumference, ask students to use a file based on the diagram on the right. Drag point B around the circumference of the circle. What do you notice about the angles at the circumference and the centre? Drag points A and C. What do you notice about the angles? If you make AOC a diameter of the circle, what do you notice about angle B? How is this a special case of the theorem that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at the circumference? Extend to other circle theorems. Proving circle theorems Ask students to develop proofs for themselves for some of the circle theorems. For example, using the first diagram, lead them to develop a proof for the special case that the angle in a semicircle is a right angle. Ask students: Which lengths are equal? What kinds of triangle can you see? Which angles are equal? How does this lead to the proof? What facts about triangles and circles do you need to use to write this proof? Ask students to use the second diagram to prove that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at the circumference. Which extra line segment has been added to the diagram to help with developing the proof? Why does this help? Ask a student to come up and share their proof with the rest of the class, then ask: Did everyone take the same steps in their proofs? Are there any differences between the proofs you produced? What facts about triangles and circles did you need to use to write this proof? As a further step, ask students to annotate their proofs to explain clearly the justification for each step. Extend to other circle theorems. While exploring constructions in this way does not constitute a proof, it does give students a sense of which properties change and which remain invariant. Angle at circumference = 60.3 Angle at centre = Drag the points A, B, C. How do the angles change? The approach here is to support students in developing proofs for themselves by providing them with diagrams that help them make a start. This column is for schools to note their own resources, e.g. textbooks, worksheets 191 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

80 Objectives Possible teaching activities Notes School resources [continued] Develop chains of logical reasoning, using correct mathematical notation and terms. Jumbled proofs Give students a copy of a proof where the statements and justifications have been jumbled up and placed in an incorrect order. Ask them to cut the statements out and sort them into a correct order so that the proof makes sense, and to annotate the diagram to show the proof. For example: BAD + BCD = 180 Angles round a point add to 360 2x + 2y = 360 Therefore opposite angles in a cyclic quadrilateral are supplementary ABC = x Angle subtended at the centre is twice that at the circumference Sum of angles in a quadrilateral is 360 x + y = 180 ADC = y Extend to other circle theorems. Security cameras A museum has a circular room that contains special items protected by security cameras mounted on the walls. The whole room needs to be protected by cameras. If each camera scans a 40 angle, how many would you need to install to cover the whole room? How many cameras would you need for different scanning angles? Circles and tangents In the diagram, the circle has lines a and b as tangents. Draw some more circles that have a and b as tangents. What is the locus of all the centres of all the circles which have the lines a and b as tangents? What is the locus of the centres of the circles which have (a) a and c, (b) b and c as tangents? What do you notice about the point where all three loci cross? Using the Internet Suggest some applets from a suitable website for students working in pairs to investigate. Each pair should make notes on what the problem is, sketch the diagram and write up their findings, justifying them as far as possible. Websites and are both good sources of applets for circle theorems and related activities. Similarly, nrich.maths.org.uk is a good source of challenging problems about circle theorems. 192 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

81 Objectives Possible teaching activities Notes School resources Inscribed circle WXYZ is a convex quadrilateral. Inside it is a circle which touches each of the sides of the quadrilateral. Prove that length WZ + length XY = length YZ + length XW. NOTE: Students may have met this problem before in Unit 11F.2. Practice and problem solving Give students a selection of problems to solve involving reasoning and proof. In some cases, preliminary exploration with DGS allows students to conjecture results or to identify properties of shapes which can then be proved. For example: Problem 1 Two smaller circles, centres B and C, touch each other externally and touch the circle centre A internally. Explore this problem using DGS, then prove it. Make the radii of the largest, middle-sized and smallest-sized circles a, b and c respectively. Then the lengths of the sides of triangle ABC are: AB = a b, AC = a c and BC = b + c. The perimeter of the triangle is: AB + BC + CA = (a b) + (a c) + (b + c) = 2a. So the perimeter of the triangle is twice the radius of the large circle, whatever the sizes of the small circles. What happens to the perimeter of triangle ABC as the two smaller circles roll around touching the rim of the bigger circle or as the two smaller circles vary in size? Problem 2 Prove that the shaded area of the semicircle is equal to the area of the inner circle. Let the radius of the inner circle be r; then its area is πr 2. The area of the semicircle is 1 2 π(2r) 2, which is 2πr 2. The percentage of the whole semicircle covered by the inner circle is 50%. 193 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

82 Objectives Possible teaching activities Notes School resources Problem 3 Two intersecting circles have a common chord AB. The 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. Explore this problem using DGS, then prove it. As C moves, so do E and F, but the common chord AB to the two circles remains fixed. Angles in the same segment are equal, so ACB = α and AEB = AFB = β (where α and β are constants). So CAF and CBE are similar. CAF = CBE = 180 α β (angles in a triangle). EAF = EBF = α + β (angles on a straight line). Since equal chords subtend equal angles at the circumference, chord EF is of constant length. Problem 5 M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P. Get students to experiment by drawing diagrams. Look at angles APM, MPD, AEM, MCD and look for cyclic quadrilaterals. Problem 6 An equilateral triangle sits on top of a square. What is the radius of the circle that circumscribes this shape? There are many ways of showing that the radius of the circle is equal to the side of the square. For example: The line OD extended is a diameter of the circle and a line of symmetry. EDC = 150 and triangle ECD is isosceles so CED = CDE = 15. Now OED = 30 so OEC = OED CED = 15 and line EC is a line of symmetry for the quadrilateral OEDC. So the radius of the circle OC is equal in length to the side of the square. 194 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

83 Objectives Possible teaching activities Notes School resources Extension: Problem 7 Three circles, two of them equal, are drawn in contact with a semicircle as shown. The radius of the semicircle is 12 cm. What is the radius of the smallest circle? This is a challenging problem. SRB is a straight line, since the normal to the common tangent at point S passes through the centres of the circles. Using Pythagoras theorem: In triangle RTP, RT 2 = (a + b) 2 (a b) 2 In triangle RTB, RT 2 = (2a b) 2 b 2 Equating the two expressions for RT 2, gives a = 2b, so b = 3, and hence the radius of the smallest circle is 3 cm. 195 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

84 Assessment Unit 11F.12 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. In the diagram ABCD is a cyclic quadrilateral. ABX, CBY, DCX and DAY are all straight lines. Calculate the values of a and b. The diagram shows three points, A, B and C, on a circle, centre O. AC is a diameter of the circle. Angle BAO is x and angle BCO is y. Explain why angle ABO must be x and angle CBO must be y. Use algebra to show that angle ABC must be 90. Two circles with centres at A and B have radii of 7 cm and 10 cm as shown in the diagram. The length of the common chord PQ is 8 cm. Calculate the length of AB. TIMSS Grade 12 In the diagram, AD is a tangent to the circle with centre O. ABC is 63 and AC is a chord of the circle. AB is 18 cm and BC is 3 cm. Calculate the values of AOC, OCA and CAD. Calculate the area of triangle ABC. Calculate the length of AC. 196 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.12 Geometry 2 Education Institute 2005

85 GRADE 11F: Algebra 4 Functions and proportionality UNIT 11F.13 8 hours About this unit This is the last of six units on number and algebra for Grade 11 foundation. It is the last in a series of four algebra units and builds on Units 11F.3, 11F.6 and 11F.10, Algebra 1, 2 and 3. 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 10 foundation or Grade 12 foundation. 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 plot straight line graphs of the form y = mx + c, and to relate the gradient and intercepts on the axes to m and c. They should recognise that the graph of y = kx 2 is a parabola through the origin. They should be able to solve a pair of simultaneous linear equations. Expectations By the end of the unit, students will solve problems involving inverse proportion. They will model situations and find exact and approximate solutions of simultaneous equations where one equation is linear and one is quadratic. They will use mathematics to model and predict the outcomes of real-world applications. Students who progress further will recognise other common examples of proportionality and use a graphics calculator to show approximate solutions to physical problems involving the intersection points of two or more graphs. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector graph plotting software such as: Autograp (see Graphmatica (free from www8.pair.com/ksoft) computers with Internet access and graph plotting software for students graphics calculators for students graph paper metre rule, small weight, fulcrum and string for each group of students Key vocabulary and technical terms Students should understand, use and spell correctly: inversely proportional, asymptote, variation, non-linear intersection, quadratic function, linear function 197 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

86 Standards for the unit Unit 11F.13 8 hours SUPPORTING STANDARDS Grade 10F and 11F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 11F and 12F standards 4 hours Inverse proportionality 4 hours Graphs of intersecting functions 11F F.5.7 Translate the statement y is proportional to x 2 into the symbolism y x 2 and into the equation y = kx 2 ; know that the graph of this equation is a parabola through the origin. Plot straight line graphs; know the meanings of gradient of the line (and be familiar with alternative wordings such as slope or rate of change of y with respect to x), 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. Interpret the solution set of the simultaneous equations E 1 and E 2, where E 1 and E 2 are the equations of two straight lines. 11F.5.16 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. 11F.5.5 Recognise some other common examples of proportional variation. 10F F F.5.15 Find exactly by algebraic means, and approximately from the points of intersection of a straight line with the graph of a quadratic function, the solution set of two simultaneous equations L 1 and Q 1, where L 1 represents a linear relation for y in terms of x, and Q 1 a quadratic function of y in terms of x. Solve physical problems modelled simultaneously by two such functions. 11F F.5.1 Use as appropriate the language of number sets from Grade 10. Use a graphics calculator, including the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs. 11F.1.2 Use mathematics to model and predict the outcomes of realworld applications; compare and contrast two or more given models of a particular situation. 198 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

87 Activities Unit 11F.13 Objectives Possible teaching activities Notes School resources 4 hours Inverse proportionality 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. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Non-linear relationships of the type xy = c Ask students to explore the following problem. A rectangular courtyard has an area of 400 m 2. Make a table of possible values for the length and width of such a courtyard and plot them on a graph. What shape is the graph? Describe the shape of the graph in as much detail as possible. What happens to the length of the courtyard as the width gets very narrow? What happens to the width of the courtyard as the length gets very short? As the length increases by 10 m, what happens to the width? How does it change? As the length doubles, what happens to the width? Ask students what other situations they can think of that are modelled by these kinds of graphs. Examples may include sharing a fixed amount between an increasing number of people or problems related to a fixed payment to do a job and the time taken to do it. Encourage students to see these relationships both as two values multiplying to make a constant (xy = c) and as one value being the constant divided by the other value (y = c/x). Practice and problem solving Give students some questions that require the use of proportional relationships, e.g. If five cats can catch five mice in five days, how many days does it take three cats to catch three mice? If a boy and a half can mow a lawn and a half in a day and a half, how many days will it take five boys to mow 20 lawns? Inverse proportion experiment Ask students to work in groups to set up the following experiment and to collect data from it. Each group will need a metre rule, small weights, a fulcrum and string. The aim is to balance the metre stick on top of the fulcrum using two weights. One weight is kept fixed and stays in one place while the other is varied and moved in order to make the metre rule balance. Students should collect data to complete the following table. Mass (g) Distance (cm) Ask students to look at the table and suggest a relationship between the two variables. They may need to be pointed towards multiplying the weight and distance together, which should give them a constant. Ask students to predict the shape of the graph and then to plot it. This column is for schools to note their own resources, e.g. textbooks, worksheets. 199 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

88 Objectives Possible teaching activities Notes School resources Different types of proportionality Ask students how they can tell whether a relationship between two numbers is a direct proportion, an inverse proportion or not a proportion at all. What do the graphs look like? What do the tables of values look like? What do the equations look like? Can you think of a practical example of each type of proportion? 4 hours Graphs of intersecting functions Find exactly by algebraic means, and approximately from the points of intersection of a straight line with the graph of a quadratic function, the solution set of two simultaneous equations L 1 and Q 1, where L 1 represents a linear relation for y in terms of x, and Q 1 a quadratic function of y in terms of x. Solve physical problems modelled simultaneously by two such functions. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Quadratic and linear functions Ask students, in pairs, to imagine a quadratic function and a linear function plotted on the same graph and to describe their graphs so that the other person can sketch it. Afterwards, ask them: Was the sketch correct? What words do you need to use? What key points do you need to make? Looking at your graph, how many intersections are there for the two functions? What are all the possibilities for the numbers of crossing points? Ask students to make sketches of all the possible numbers of solutions for solving linear and quadratic functions simultaneously. Prepare some function cards containing linear functions on one colour of card and quadratic functions on another. Ask students to pick two cards at random, one of each colour, and to use a graphics calculator or graph plotting software to find an approximate solution to the pair of simultaneous equations they have. Can you find a pair of cards with no solutions? One solution? Two solutions? Solving equations algebraically Ask students how they would solve the pair of simultaneous equations y = x 2 + 5x and y = 3x algebraically to get an exact answer. What do you know about the x- and y-values of the intersection points of the curve and the line? How does this help you to solve the pair of equations? How does this lead to you to one, two or zero solutions? Practice Give a practice exercise involving solving simultaneous equations algebraically. The website mathsnet.net/asa2/2004/c12qintersect.html contains an interactive sketch of quadratic and linear functions solved simultaneously. Students should be able to realise that, as the two equations are usually given explicitly, they can equate them and that this gives a quadratic which will give one, two or zero solutions. The website mathsnet.net/asa2/2004/c12qintersect_2.html contains a useful sorting activity on the steps used in solving such pairs of simultaneous equations algebraically. 200 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

89 Objectives Possible teaching activities Notes School resources Solving problems using simultaneous equations Give students a range of problems that require the formulation and solution of simultaneous equations. For example: Food can A cylindrical food can is made from a sheet of metal. It has a surface area of 680 cm 2 and the net of the can is shown in the diagram. The full width of the sheet of metal the can is made from is 25.7 cm. Write down a pair of simultaneous equations for the radius and height of the can. Solve this pair of equations simultaneously and find the volume of the can. Flower bed An ornamental flower bed is designed as shown in the diagram. Each area of planting is x m by x m square and the dividing path is y m wide. The total area of planting is 40 m 2 and the total area of path is 12.6 m 2. Write down an equation for the area of the whole flower bed. Write down an equation for the planted area. 201 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

90 Assessment Unit 11F.13 Examples of assessment tasks and questions Notes School resources Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Three people working flat out complete a job in sixteen hours. How many hours would it take eight people to do the same job? Explain any assumptions you have made. Look at the graphs. a. One of the graphs shows the equation y = kx x 2 (k is a constant). Which graph is it? b. One of the graphs shows the equation y = k/x, where k is a positive constant. Which graph is it? The average speed for a fixed-distance journey is inversely proportional to the time taken to complete the journey. A family travels in Europe by car. They travel exactly half their journey in 2 hours, then stop for lunch for 1 hour, and then take 3 hours over the second half of the journey. How were the average speeds related on each part of the journey? If the average speed for the first half of the journey was 72 kilometres per hour, what was the average speed for the whole journey? Explain why the function y = k/x cannot be defined on the domain set. What is the largest domain the function can be defined on? Sketch the graph of the function for this domain. Does the function have a greatest or least value? Is there anywhere where the function increases? 202 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.13 Algebra 4 Education Institute 2005

91 GRADE 11F: Geometry and measures 4 Compound measure and circle problems UNIT 11F.14 4 hours About this unit This is the last of four units on geometry and measures for Grade 11 foundation. It builds on the work in Units 11F.4, 11F.7 and 11F.9, Geometry and measures 1, 2 and 3. 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 10 foundation or Grade 12 foundation. 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 be able to use formulae to calculate: the circumference and area of a circle; the perimeter and area of any triangle, or trapezium, parallelogram or quadrilateral with perpendicular diagonals; the surface area and volume of a right prism, cylinder, square-based pyramid and cone; and the volume of a sphere. They should be familiar with SI units and be able to use them to calculate rates, density and average speed. Expectations By the end of the unit, students will continue to use SI units and a range of measures to solve problems, including radian measure to calculate sector areas and arc lengths and compound measures. 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 solve a range of problems involving compound measures, using appropriate units and dimensions. Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector computers with Internet access for students plain paper, compasses, string, scissors for each student poster and modelling materials plastic football and marker pens, orange and fruit knife atlases and globe Key vocabulary and technical terms Students should understand, use and spell correctly: radian, sector, arc, centre, radius, circumference, diameter, cone bearings, longitude, latitude, great circle, hemisphere, Equator, nautical mile compound measure, speed, acceleration, rate, average speed 203 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.14 Geometry and measures 4 Education Institute 2005

92 Standards for the unit Unit 11F.14 4 hours SUPPORTING STANDARDS Grade 10F standards CORE STANDARDS Grade 11F standards EXTENSION STANDARDS Grade 12F standards 2 hours Circle problems 2 hours Measures 10F.7.1 Use formulae to calculate: the circumference and area of a circle; the perimeter and area of any triangle, or trapezium, parallelogram or quadrilateral with perpendicular diagonals; the surface area and volume of a right prism, cylinder, square-based pyramid and cone; and the volume of a sphere. 11F.6.13 Use radian measure to calculate sector areas and arc lengths. 11F.7.2 Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth s surface. 11F.7.1 Calculate lengths, areas and volumes of geometrical shapes. 10F.7.3 Work with SI units and compound measures: rates such as cost per litre, kilometres per litre, litres per kilometre; and average speed and density, including population density (number of people per unit area). 11F F.1.3 Work with SI units and compound measures including density, average speed and acceleration, measures of rate, and population density (number of people per unit area), using appropriate units and dimensions. Identify and use interconnections between mathematical topics. 12F.9.3 Solve problems involving compound measures, using appropriate SI units and dimensions. 11F.1.2 Use mathematics to model and predict the outcomes of realworld applications; compare and contrast two or more given models of a particular situation. 11F.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems. 204 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.14 Geometry and measures 4 Education Institute 2005

93 Activities Unit 11F.14 Objectives Possible teaching activities Notes School resources 2 hours Circle problems Use radian measure to calculate sector areas and arc lengths. Use bearings, latitude, longitude and great circles to solve problems relating to position, distance and displacement on the Earth s surface. Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation. Solve routine and non-routine problems in a range of mathematical and other contexts, including openended and closed problems. What is a radian? Ask students to draw a large circle on a piece of A4 paper, to mark the centre and to draw in a radius. Now ask them to cut a piece of string equal to the length of the circumference of the circle. Explain that if there were no such thing as degrees we could measure an angle by drawing a circle centred on the angle and counting the number of radiuses of the circle that fit along the arc for the angle. Students could verify and then explain why the size of the angle is independent of the radius of the circle. Now ask students to take the piece of string and cut off pieces that are the same length as the radius of their circle. How many pieces of string do you get? What special number is this? Why? Now ask students to stick one of their radius-length pieces of string onto the circumference of their circle. Use this to lead into a discussion of the definition of a radian and that there are 2π radians in a circle. Arcs and sectors in radians Derive the formulae for arc length and the area of a sector using radians as the measure of angle. Give students some practice in calculating these. Big Ben The clock face on the Clock Tower in London s Palace of Westminster, known as Big Ben, has a diameter of 23 feet. The minute hand of the clock is 14 feet long. Given that 1 foot is cm, how long is the minute hand in metres? What distance does the tip of the minute hand travel in one hour? How far does it travel in one minute? Do you think you could see it move? What is the area of the sector of the clock between the numerals 1 and 2? Great circles Ask students to use the Internet to find out about great circles and their uses. Some websites will allow them to calculate the great-circle distance between two places, and others use latitude and longitude, place name or a map to specify location. Use the applet Great circle (nlvm.usu.edu/en/nav/vlibrary.html) to assist their understanding. Ask students to present the information they have found as a poster or as a model. This column is for schools to note their own resources, e.g. textbooks, worksheets. 205 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.14 Geometry and measures 4 Education Institute 2005

94 Objectives Possible teaching activities Notes School resources Modelling the Earth Use a plastic football to make a model of the Earth. Use an atlas to mark on the North and South Poles and the locations of Bahrain and Tokyo. Ask students to use string to find the shortest distance between the two locations. Repeat this activity using an orange to represent the Earth: cut the orange along the line of greatest distance and demonstrate that this splits the orange into two hemispheres. Longitude and latitude treasure hunt Ask students to work in pairs to design a treasure hunt using longitude and latitude to identify places on a globe. Students should exchange treasure hunts with another pair and try to follow one another s clues. Encourage students to use clues that have the following form: Cairo is 30 2 N, E. What are the latitude and longitude of the position which is a reflection of Cairo s position in the equatorial plane? Jakarta is 6 16 S, E. What are the latitude and longitude of the position starting at Jakarta and following a rotation 60 W followed by a rotation 70 N? What position do you reach if you start at Ankara, which is N, E, and move 40 E? Calculate the distance along parts of your route. Practice and follow up What is the latitude and longitude of Palm Tree Island? How does the circumference of the circles of latitude vary for different latitudes? Work out the answers in nautical miles. Plan a round-the-world cruise using latitude and longitude for positions and great-circle routes. Palm Tree Island, Doha 2 hours Measures Calculate lengths, areas and volumes of geometrical shapes. [continued] Length, area and volume Remind students about the formulae for calculating the perimeters and areas of rectangles, the areas of triangles, parallelograms and trapeziums, and the volumes and surface areas of cubes, prisms, pyramids, cones and spheres. Give practice in calculating these. Take the opportunity to revise how to round answers to a given number of significant figures. 206 Qatar mathematics scheme of work Grade 11 foundation Unit 11F.14 Geometry and measures 4 Education Institute 2005

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