1.1 sequences get in line - pearson global schools · 2016-06-14 · 18 get in line 1.1 sequences...

18 Get in line

1.1 Sequences

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, defi nitions and derived properties

• Use a ruler and protractor to measure and draw angles, including refl ex 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 defi nition of a circle and the names of its parts

• Explain how to fi nd, 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

Website links

• OpenerOnline angle puzzles

• 2.5 Geometry resources, including interactive explanations

2 Get in line

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Opener 19

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 simplifi ed 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 auto-marked tests are provided to match this unit.

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20 Get in line

2.1 WGM pages to come

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Sequences 21

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22 Get in line

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, defi nitions and derived properties

Starter (1) Oral and mental objective

Display this table and ask pupils to fi nd 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? (angles BAC, ACB, CBA) Exterior angle? (angle BCD)

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 fi nd 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

2.2 Angles and proof

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

A C

B

D

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– 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 fi nding the sums of the angles in the two triangles formed. Q5

– Explain the difference between conventions, defi nitions 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 defi ned? 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 defi nition of the shape? Q7

Activity APupils make up their own triangles and give the sizes of two of the interior angles. They challenge their partners to fi nd the missing interior and exterior angles.

Activity BPupils investigate the interior angles in a triangle using dynamic geometry software.

PlenaryDisplay a right-angled triangle.

Ask pupils how they would prove that a + b = 90°.

HomeworkHomework Book section 2.2.

Challenging homework: Pupils investigate fi nding 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 angleb) 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) Defi nition d) Convention

2.2 Angles and proof 23

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 diffi culties

Pupils can fi nd moving to formal proof diffi cult so encourage the use of symbols before moving onto letters.

LiveText resources

Explanations

Booster

Extra questions

Worked solutionsUNCORRECTED PROOF

24 Get in line

Objectives

• Use a ruler and protractor to measure and draw angles, including refl ex 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)


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.


Display angles on the board and ask pupils to identify whether they are acute, obtuse or refl ex 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 fi rst? 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 fi rst. 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

2.3 Constructing triangles

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

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– 4 Construct a right-angled triangle using compasses

Display a right-angled triangle. Which side is the hypotenuse? How can you draw a right-angled 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) refl ex c) refl ex 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.

2.3 Constructing triangles 25

Related topics

Discussion points


Encourage pupils to check their measurements using a ruler as sometimes the compass can slip.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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26 Get in line

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



Display the following target board and ask pupils to fi nd complements to 360.

240 180 270 143

172 84 140 138

145 135 78 90

294 265 127 232


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 specifi c quadrilateral and fi nds 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 fi ndings of each group with the rest of the class and summarise the fi ndings on the board. Q1–5

Resources

Starter (2): mini whiteboards, paper, scissors

Main: poster paper

Intervention

Functional skills


Framework 2008 ref

1.3, Y8 1.4, Y8 4.1, Y9 1.2

PoS 2008 refUNCORRECTED PROOF

Display this shape and model how to fi nd the missing angles.

During each step of their working, ask pupils to explain their reasoning and show this on the board. Q6–10

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


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 of pairs of

parallel sides

Lines of symmetry

0 1 2 4

0 kite, arrowhead

1 isosceles trapezium

2 parallelogram rectanglerhombus

square

4 b) Parallelogramc) 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.

2.4 Special quadrilaterals 27

Related topics

Art and Design Technology.

Discussion points

Is a rectangle a square? Is a parallelogram a rhombus?


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

a

c

b

120°35°

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28 Get in line

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 defi nition of, and the names of parts of a circle


Introduce the term ‘bisect’. Practise fi nding halves of numbers and measures, for example 5 cm, 3.3 cm, 45°.


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

Resources

Starter (1): mini whiteboards

Main: compasses, rulers, protractors

Activity A: dynamic geometry software (optional)

Intervention

Functional skills


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

UNCORRECTED PROOF

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 fi nd this easier if they have done Activities A and B.

Homework


Challenging homework: Pupils could make other constructions such as the centroid of a triangle, or use perpendicular bisectors to fi nd the centre of a circle.

Answers1 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.



8 a) b) Circles correctly drawn.c) It is a rhombus.

2.5 More constructions 29

Related topics

Loci


Encourage pupils to check their measurements using a ruler as sometimes the compasses can slip.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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30 Get in line

Objectives

• Explain how to fi nd, calculate and use: the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons; the interior and exterior angles of regular polygons


2.6 Angles in polygons


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.


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 fi nd 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?

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

UNCORRECTED PROOF

Sequences 31

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


Challenging homework: Pupils could fi nd 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

6 720° 120° 360° 60°

regular octagon

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


LiveText resources

Explanations

Booster

Extra questions

Worked solutions

UNCORRECTED PROOF

32 Get in line

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 benefi cial 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

(1 mark per triangle)

Puzzle time

6 cm

8 cm

6 cm

8 cm

6 cm

8 cm

UNCORRECTED PROOF

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 defi ned in the QCA guidelines:

• Select the mathematical information to use

• Use appropriate mathematical procedures

• Find results and solutions

Puzzle time 33

UNCORRECTED PROOF

1.1 sequences get in line - pearson global schools · 2016-06-14 · 18 get in line 1.1 sequences...

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