21.1 Sequences Get in line Unit objectives Understand a proof that the angle sum of a triangle is 180 and of a quadrilateral is 360 ; and the exterior angle of a triangle is equal to the sum of the two interior opposite angles Distinguish between conventions, definitions and derived properties Use a ruler and protractor to measure and draw angles, including reflex angles, to the nearest degree; and construct a triangle, given two sides and the included angle (SAS) or two angles and the included side (ASA) Use straight edge and compasses to construct triangles, given right angle, hypotenuse and side (RHS) Solve geometrical problems using side and angle properties of equilateral, isosceles and right-angled triangles and special quadrilaterals, explaining reasoning with diagrams and text; classify quadrilaterals by their geometrical properties Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons Use straight edge and compasses to construct the mid-point and perpendicular bisector of a line segment; the bisector of an angle; the perpendicular from a point to a line; the perpendicular from a point on a line Know the definition of a circle and the names of its parts Explain how to find, calculate and use the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons; and the interior and exterior angles of regular polygons 18 Get in line Website links Opener Online angle puzzles 2.5 Geometry resources, including interactive explanations
Notes on the context Recreational maths (puzzles and games that relate to maths) can intrigue and inspire those who are not naturally drawn to maths as a subject. The Englishman Henry Dudeney and the American Sam Loyd, who worked on and published puzzles at much the same time, did not have strong mathematical backgrounds but both found puzzles irresistible. Dudeney and Loyd collaborated for a time, but their working relationship broke down when Dudeney accused Loyd of stealing his ideas and publishing them as his own. Dudeney s original instructions for solving the Haberdasher s problem included constructions using ruler and compasses, e.g. for the bisection of two sides of the triangle. The base of the triangle is cut in the approximate ratio 0.982 : 2 : 1.018. A simplified solution is given here: Bisect AC; bisect BC. Roughly divide AB into three in the ratio 0.982 : 2 : 1.018. Draw the lines as shown lines meet at right angles inside the triangle. Then rearrange the pieces. For a range of other fun dissection puzzles, which can be downloaded as resource sheets, please visit the relevant unit section at www.heinemann.co.uk/ hotlinks. Discussion points What mathematical skills are used in activities A and B? Activity A a) b) Activity B a) b) c) Answers to diagnostic questions 1 Pupil s line 6.3 cm long 2 31 3 Pupil s angle of 87, labelled acute 4 a) rectangle b) equilateral triangle c) square d) scalene triangle 5 Square, rectangle, trapezium, parallelogram, rhombus, kite, arrowhead LiveText resources Paper planes Use it! Games Quizzes Get your brain in Gear Audio glossary Skills bank Extra questions for each lesson (with answers) Worked solutions for some questions Boosters Level Up Maths Online Assessment The Online Assessment service helps identify pupils competencies and weaknesses. It provides levelled feedback and teaching plans to match. Diagnostic automarked tests are provided to match this unit. Opener 19
2.1 WGM pages to come 20 Get in line
Sequences 21
2.2 Angles and proof Objectives Understand a proof that: the sum of the angles of a triangle is 180 ; and of a quadrilateral is 360 ; and the exterior angle of a triangle is equal to the sum of the two interior opposite angles Distinguish between conventions, definitions and derived properties Starter (1) Oral and mental objective Display this table and ask pupils to find complements to 180. 120 40 170 43 72 84 140 38 145 135 78 90 94 165 127 32 Starter (2) Introducing the lesson topic Recap alternate and corresponding angles on parallel lines. Using mini whiteboards, ask pupils to draw a pair of parallel lines with a transversal. Ask pupils to mark a pair of corresponding angles and a pair of alternate angles. Main lesson Explain that pupils will be using what they know about angles on parallel lines to prove that the interior angles in a triangle sum to 180. 1 Interior and exterior angles Display this diagram. Which of these angles is an interior angle? B (angles BAC, ACB, CBA) Exterior angle? (angle BCD) A C D What is the sum of the interior angles in a triangle? Angle BCA is 50. Calculate angle BCD. (130 ) Repeat with other values of angle BCA. Give other interior angles in the triangle to check pupils are able to find missing interior and exterior angles. Repeat for a quadrilateral. Q1 3 2 Proof of sum of interior angles in a triangle Display this diagram. Ask pupils to copy the diagram on mini whiteboards and label the other angles which are equal to the circle and triangle. Lead pupils through the proof that if angles on a straight line add up to 180, then the angles in the triangle must also sum to 180. Q4, 6 22 Get in line Resources Starter (2), Main: mini whiteboards Activity B: dynamic geometry software Intervention Functional skills Make an initial model of a situation using suitable forms of representation Framework 2008 ref 1.3, Y8 1.2, Y8 4.1, Y9 4.1, Y9 4.3 PoS 2008 ref
3 Proof of sum of interior angles in a quadrilateral Model how pupils can prove that the sum of the exterior angles in a quadrilateral is 360 by drawing a diagonal from a vertex to the opposite vertex, and finding the sums of the angles in the two triangles formed. Q5 Explain the difference between conventions, definitions and derived properties. Many pupils struggle with this so try to provide as many examples as possible and ask pupils to suggest their own examples. Display a simple shape such as a square. How could this shape be defined? What conventions are used to show that the angles are 90 and the sides are the same length? What derived properties can be deduced from the definition of the shape? Q7 Activity A Pupils make up their own triangles and give the sizes of two of the interior angles. They challenge their partners to find the missing interior and exterior angles. Activity B Pupils investigate the interior angles in a triangle using dynamic geometry software. Plenary Display a right-angled triangle. Ask pupils how they would prove that a + b = 90. Homework Homework Book section 2.2. Challenging homework: Pupils investigate finding the proof that the sum of the exterior angles of a triangle is 360. Answers 1 p = 100, 2 a) i) An exterior angle ii) An interior angle iii) An exterior angle b) 75 i) 96 ii) 82 iii) 63 3 a) x = 91, interior angles in quadrilateral sum to 360 ; y = 89, angles on a straight line add up to 180. b) s = 55, t = 55, angles in a triangle sum to 180, isosceles triangle has two equal angles; u = 125, angles on a straight line add up to 180 or exterior angle of a triangle equals the sum of the two interior opposite angles. c) q = 75, angles on a straight line add up to 180 ; p = 47, angles in a triangle sum to 180, or exterior angle of a triangle equals the sum of the two interior opposite angles. d) d = 88, interior angles in quadrilateral sum to 360 ; e = 82, angles on a straight line add up to 180. 4 Angle x is equal to angle a because they are alternate angles. Angle y is equal to angle c because they are alternate angles. x + b + y = 180 because they lie on a straight line. Since x = a and y = c, a + b + c = x + b + y. This proves that angles in a triangle sum to 180 5 a + b + c = 180 because angles in a triangle sum to 180. d + e + f = 180 because angles in a triangle sum to 180. Therefore (a + b + c) + (d + e + f ) = 360. 6 a + b + c = 180 because angles in a triangle sum to 180. c + x = 180 because they lie on a straight line. a + b + c = c + x 7 a) Derived property b) Convention c) Definition d) Convention Related topics Symmetry and art Discussion points Discuss what constitutes a proof and the difference between demonstrating a rule works and proving that the rule is always true. Common difficulties Pupils can find moving to formal proof difficult so encourage the use of symbols before moving onto letters. LiveText resources Explanations Booster Extra questions Worked solutions 2.2 Angles and proof 23
2.3 Constructing triangles Objectives Use a ruler and protractor to measure and draw angles, including reflex angles, to the nearest degree Construct a triangle given two sides and the included angle (SAS) or two angles and the included side (ASA) Use straight edge and compasses to construct a triangle, given right angle, hypotenuse and side (RHS) Starter (1) Oral and mental objective Ask pupils to visualise a square piece of paper. I fold it across one of the diagonals. What shape is made? What are the angles in the shape? I fold the resulting shape in half. What shape do I get? What angles are in the new shape? Ask pupils to explain their reasoning. Starter (2) Introducing the lesson topic Display angles on the board and ask pupils to identify whether they are acute, obtuse or reflex angles. Ask pupils to estimate the size of the angles. Ask pupils to draw an acute angle of 72. Pupils check their angle drawing with their partner. Main lesson Explain that pupils will be constructing triangles using a protractor and a ruler and also compasses and a ruler. They should already have done this, so some of this lesson will be revision. 1 Construct a triangle given two sides and an angle (SAS) Recap on how to draw a triangle given two sides and an angle using a protractor and a ruler. What will you measure and draw first? Q1 2 2 Construct a triangle given two angles and a side (ASA) How do I draw a triangle given two angles and a side using a protractor and a ruler? Q3 4 3 Construct a triangle given three sides (SSS) I know the lengths of all three sides of a triangle. How do I use compasses and a ruler to draw the triangle? Model how to draw a triangle, for example with sides 8 cm, 5 cm, 6 cm. Advise pupils to draw the longest side first. Ensure that they can use compasses correctly. Q6 7 Display a straight line. How do I construct a line perpendicular to this line? Check that pupils know how to do this. Q8 24 Get in line Resources Starter (2): compasses, ruler, protractor Intervention Functional skills Use appropriate mathematical procedures Framework 2008 ref 1.3, Y8 1.2, Y8 4.3, Y9 4.3 PoS 2008 ref
4 Construct a right-angled triangle using compasses Display a right-angled triangle. Which side is the hypotenuse? How can you draw a rightangled triangle when you know the length of the hypotenuse and one of the other sides? Model how to use compasses and a ruler to do this. For example draw a sketch of a right-angled triangle then model how to draw the right-angled triangle with a hypotenuse of 15 cm and one side 9 cm. Repeat with another triangle if appropriate. What is the length of the unknown side? Q5, 9 11 Activity A Pupils practise drawing a triangle using a protractor and ruler and then describe it for their partner to draw. Activity B Pupils practise drawing a right-angled triangle using compasses and a ruler and then describe it for their partner to draw. Plenary Ask pupils which triangles are impossible to draw. Give them two minutes to discuss in small groups and then share their answers with the rest of the class. Write a selection of answers on the board. Homework Homework Book section 2.3. Challenging homework: Pupils construct nets using compasses and a straight edge. Answers 1 Correct angles drawn. a) obtuse b) reflex c) reflex d) obtuse 2 Correct triangles drawn. 3 Accurate drawing of triangles. 4 b) 10 + 11 = 21 m 5 a) b b) d c) i d) j 6 Accurate drawing of triangle. 7 Accurate drawing of triangle. 8 Perpendicular line drawn. 9 a) Correct scale drawing. b) 6 m 10 a) Correct scale drawing. b) 3.9 m 11 The two shorter sides are 5 cm and 3 cm. These add up to 8 cm, which is shorter than the third side 9 cm. Therefore the shorter sides will never meet. Related topics Discussion points Common difficulties Encourage pupils to check their measurements using a ruler as sometimes the compass can slip. LiveText resources Explanations Booster Extra questions Worked solutions 2.3 Constructing triangles 25
2.4 Special quadrilaterals Objectives Begin to identify and use angle, side and symmetry properties of triangles and quadrilaterals Solve geometrical problems using side and angle properties of equilateral, isosceles and right-angled triangles and special quadrilaterals; explaining reasoning with diagrams and text; classifying quadrilaterals by their geometric properties Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons Starter (1) Oral and mental objective Display the following target board and ask pupils to find complements to 360. 240 180 270 143 172 84 140 138 145 135 78 90 294 265 127 232 Starter (2) Introducing the lesson topic Ask pupils to draw a rectangle on a piece of paper and cut it out. Pupils draw and measure the diagonals of the rectangle. What do you notice about where the diagonals cross? (bisect each other) In pairs, ask pupils to write three sentences to describe the rectangle. Explain that they can comment on things like the sides, angles and symmetry. Take feedback about the sentences they have written. Write a selection on the board. Main lesson Explain that pupils will be investigating the properties of special quadrilaterals. 1 Special quadrilaterals Display a rectangle, square, parallelogram, rhombus, isosceles trapezium, kite and arrowhead and ask pupils to name the ones that they already know. Ask pupils to work in groups each group focuses on a specific quadrilateral and finds its properties. Each group could make a poster of the properties of their shape and this could be displayed during the lesson for the class to use. Share the findings of each group with the rest of the class and summarise the findings on the board. Q1 5 26 Get in line Resources Starter (2): mini whiteboards, paper, scissors Main: poster paper Intervention Functional skills Use appropriate mathematical procedures Framework 2008 ref 1.3, Y8 1.4, Y8 4.1, Y9 1.2 PoS 2008 ref
Display this shape and model how to find the missing angles. During each step of their working, ask pupils to explain their reasoning and show 35 this on the board. Q6 10 b a 120 c Activity A Pupils work in pairs, using the properties of quadrilaterals to identify the shape. Activity B In this activity pupils set problems for their partner to solve within a parallelogram. Plenary Give pupils the following description: I am a special quadrilateral. I have one line of symmetry and two pairs of equal sides. I have no parallel lines. Which special quadrilateral am I? (kite) Repeat with other descriptions. Homework Homework Book section 2.4. Challenging homework: Pupils could identify impossible quadrilaterals if sides and angles are given. Answers 1 Yes a square is a rectangle with all sides of equal length. 2 C 3 Number Lines of symmetry of pairs of parallel sides 0 1 2 4 0 kite, arrowhead 1 isosceles trapezium 2 parallelogram rectangle rhombus square 4 b) Parallelogram c) Opposite sides are equal and parallel; diagonals bisect each other; rotation symmetry of order 2. 5 a) Rhombus b) A, C 6 a = 60, b = 30, c = 60 7 x = z = 140, y = 40 8 a) TUV = 45 b) TVU = 105 c) SVU = 150 9 ABE = 180 90 72 = 18. CBD = 180 90 56 = 34. (Angles in a triangle sum to 180.) ABC = 90, therefore EBD = 90 18 34 = 38. There are other valid approaches. 10 a) FAB = 65 (Opposite angles in a parallelogram are equal.) b) ABE = 70 (Alternate angles are equal.) c) CBE = 110 (Angles on a straight line sum to 180.) d) BCD = 115 (Angles in a quadrilateral sum to 360.) There are other valid approaches. Related topics Art and Design Technology. Discussion points Is a rectangle a square? Is a parallelogram a rhombus? Common difficulties When pupils are asked to describe the properties it is useful to display key words and a list of what to comment on when describing their shapes. LiveText resources Explanations Booster Extra questions Worked solutions 2.4 Special quadrilaterals 27
2.5 More constructions Objectives Use straight edge and compasses to construct: the mid-point and perpendicular bisector of a line segment; the bisector of an angle; the perpendicular from a point to a line; the perpendicular from a point on a line Know the definition of, and the names of parts of a circle Starter (1) Oral and mental objective Introduce the term bisect. Practise finding halves of numbers and measures, for example 5 cm, 3.3 cm, 45. Starter (2) Introducing the lesson topic Ask pupils to draw a circle on mini whiteboards. Ask them to draw and label the diameter, radius, circumference, chord, arc, sector, tangent. Check pupils drawings and identify the parts of a circle on the board. Main lesson What does the term perpendicular mean? Check that pupils know. Explain that pupils will not be using a protractor to measure angles but that they will be drawing perpendicular lines using compasses and a ruler only. Most of this is revision of earlier work. 1 Construct the perpendicular bisector of a line segment How do you draw the perpendicular bisector of a line segment? Take instructions from pupils to check that they know how to do this remind them if necessary. Also check that they keep the compasses rigid while drawing the perpendicular bisector. Q1 3 2 Construct the angle bisector How do you draw the bisector of an angle using compasses only? Remind pupils, if necessary (they should have done this in earlier work), and give them an opportunity to practise. Pupils can check they have bisected the angle accurately by checking with a protractor. Q4 3 Construct the perpendicular from a point on a line segment How do you construct the perpendicular from a point on a line segment? Take instructions from pupils to check that they know how to do this remind them if necessary. Q6, 7 4 Construct the perpendicular from a point to a line segment How do you construct the perpendicular from a point to a line segment? Take instructions from pupils to check that they know how to do this remind them if necessary. Q5, 8 28 Get in line Resources Starter (1): mini whiteboards Main: compasses, rulers, protractors Activity A: dynamic geometry software (optional) Intervention Functional skills Use appropriate mathematical procedures Framework 2008 ref 1.3, Y8 1.2, Y9 1.1, Y9 4.1, Y8 4.3 PoS 2008 ref Website links www.heinemann.co.uk/ hotlinks
Activity A Pupils practise drawing the perpendicular bisector for a triangle in a circle. If available, dynamic geometry software is useful for this activity. In a triangle, the perpendicular bisectors meet at the circumcentre of the triangle. Activity B Pupils draw polygons within circles and investigate where the perpendicular bisectors of the sides intersect. Plenary Ask pupils how you can draw a circle whose circumference passes through each vertex of a triangle. Give them a few minutes to discuss their ideas in groups and then report back to the class. Write a summary on the board. Pupils will find this easier if they have done Activities A and B. Homework Homework Book section 2.5. Challenging homework: Pupils could make other constructions such as the centroid of a triangle, or use perpendicular bisectors to find the centre of a circle. Answers 1 Perpendicular bisectors correctly drawn. 2 b) Perpendicular bisector correctly drawn. c) It is an equal distance from both houses. 3 Circle with radius, diameter, chord, arc, tangent, circumference correctly labelled. 4 Perpendicular bisectors correctly drawn. 5 Perpendicular correctly drawn. 6 Perpendicular correctly drawn. 7 Perpendicular correctly drawn. 8 a) b) Circles correctly drawn. c) It is a rhombus. Related topics Loci Common difficulties Encourage pupils to check their measurements using a ruler as sometimes the compasses can slip. LiveText resources Explanations Booster Extra questions Worked solutions 2.5 More constructions 29
2.6 Angles in polygons Objectives Explain how to find, calculate and use: the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons; the interior and exterior angles of regular polygons Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons Starter (1) Oral and mental objective Ask pupils to add and subtract pairs of numbers, for example the answer is 149 what is the question? Ask pupils to list pairs of numbers that you can add to make 149. Repeat for numbers such as 8.6, 0.4, 0.12. Starter (2) Introducing the lesson topic Recap the sum of the interior angles in a triangle. Which of these sets of angles are angles in a triangle? Explain your reasoning. A 36, 72, 93 B 59, 73, 48 Two angles in a triangle are 48 and 87. Calculate the missing angle. Main lesson 1 Proof of sum of interior angles in a quadrilateral Remind pupils that they proved that the sum of angles in a quadrilateral is 360. Display an irregular quadrilateral. How can you split it up into triangles? Label the angles in one triangle a, b and c and in the other triangle d, e and f. Show how a + b + c = 180 and d + e + f = 180 and therefore angles in a quadrilateral must sum to 360 Q1 2 Sum of the interior angles in polygons Display this table: Shape Number of sides Number of triangles Sum of interior angles triangle 3 1 1 180 = 180 quadrilateral 4 2 2 180 = 360 pentagon hexagon Ask pupils to complete the missing values. For an n-sided polygon, how would you find the number of triangles? (n 2) Sum of interior angles? ((n 2) 180) Q2 4 3 Sum of the exterior angles in polygons Display a quadrilateral. What is an exterior angle? How would you work out the sum of the exterior angles in a polygon? What is the sum? 30 Get in line Resources Activity A: materials for poster making Intervention Functional skills Make an initial model of a situation using suitable forms of representation Framework 2008 ref 1.3, Y8 1.2, Y9 1.2, Y9 4.1 PoS 2008 ref
Explain that in a regular polygon all the sides have the same length and the angles are equal. How would you calculate one of the interior angles in a regular hexagon? (720 6 = 120 ) What is the size of one of the exterior angles? (60 ) Discuss both of the following methods: Method (1): 360 6 = 60 Method (2) 180 interior angle Q5 11 Activity A Pupils make a poster explaining what they know about interior and exterior angles in polygons. Activity B Pupils try to explain which regular polygons tessellate by looking at their interior angles. Plenary Ask pupils if it is possible to draw a polygon whose interior angle sum is 1400. Give them a short time to discuss this in small groups and report back to the class. Repeat for other values. Homework Homework Book section 2.6. Challenging homework: Pupils could find examples of real-life regular polygons, and calculate interior and exterior angles. Answers 1 a) Split the shape into two triangles. b) Spit the shape into three triangles. 2 a) Find the sum of the interior angles by dividing the pentagon into three triangles, then divide by 5. b) Subtract the interior angle from 180. 3 a) i) 360 ii) 540 iii) 720 b) 1440 4 The interior and exterior angles lie on a straight line. Angles that form a straight line sum to 180. 5 b) 360 c) 360 d) Sum of exterior angles is always 360. 6 a) 60 b) 120 7 Regular polygon Number of sides Sum of interior angles Size of each interior angle Sum of exterior angles Size of each exterior angle equilateral triangle 3 180 60 360 120 square 4 360 90 360 90 regular pentagon 5 540 108 360 72 regular hexagon regular octagon 6 720 120 360 60 8 1080 135 360 45 8 a (n 2) 180 b) Interior 157.5, exterior 22.5 9 a) i) 20 ii) 162 b) 15 10 No. The sum of the interior angles in a multiple of 180 and 1300 is not divisible by 180. 11 a) 135 b) 45 c) 22.5 Related topics Art and design, design technology, ICT Common difficulties LiveText resources Explanations Booster Extra questions Worked solutions Sequences 31
Puzzle time Notes on plenary activities The activities cover a range of missing angle problems. It would be useful to discuss pupil methods for the latter questions, particularly activities 8 and 9. Emphasise that surplus details are not given in these types of problems all information given will and should be used to reach a solution. What does the arrow notation represent? How can this be used to solve problems? It would be beneficial to summarise the learning in this unit by highlighting the important angle facts producing a checklist for angle problems could also be useful. Solutions to the activities 1 a = 135 2 b = 30 3 c = 142, d = 65 4 e = 71 5 f = 104, g = 96, h = 84 6 i = 119, j = 61 7 k = 234 8 l = 45, m = 65, n = 70 9 o = 105, p = 75, q = 105 10 r = 170 Number grid: Answers to practice SATs-style questions 1 a) Angles on a straight line sum to 180. 180 70 = 110, so Sally is correct. b) a = 45 (1 mark each) 2 a = 40, b = 140, c = 20 (1 mark each) 3 a) Angle BCD = 105 b) Angle BAD = 75 (1 mark each) 4 6 cm 6 cm 8 cm 8 cm 8 cm 6 cm (1 mark per triangle) 32 Get in line
5 a) 3y = 90, so y = 30 (2 marks) b) 2x = 30, so x = 15 (2 marks) 6 a) ABCD: interior angles sum to 360, so angle ADC = 96 and angle EDC = 48 (2 marks) b) Angle DEB = 132 (1 mark) c) DAE is an isosceles triangle: angle DAE = 84, angle ADE = 48 and angle AED = 48 (1 mark) 7 a) s = 32 b) t = 56 (2 marks each) Functional skills The plenary activity practises the following functional skills defined in the QCA guidelines: Select the mathematical information to use Use appropriate mathematical procedures Find results and solutions Puzzle time 33