Turning triangles task 2

Transcription

1 Q. Turning triangles task (a) This regular pentagon is made from 5 isosceles triangles that fit together around a point. The triangles fit with no gaps and no overlaps. Work out the angles in one of the triangles. (b) A regular decagon can be made from 0 isosceles triangles that fit together around a point with no gaps and no overlaps. Work out the angles in one of these triangles. (c) All regular polygons can be made from isosceles triangles that fit together around a point with no gaps and no overlaps. Only 2 of these regular polygons have isosceles triangles in which all the angles are whole numbers, and all the angles are whole numbers than or equal to 0. How many sides do these polygons have, and how can you be certain that there are no more than 2 of these polygons? Turning triangles task 2 (a) Isosceles triangles can fit together around a point in a different way to make windmill patterns. The triangles fit with no gaps and no overlap. Work out the angles in the triangle. Page of 2

2 (b) This windmill pattern has been made into a regular pentagon by drawing five extra triangles. Work out the angles in each triangle. (c) Other windmill patterns can be made into regular polygons in the same way by drawing extra triangles, with angles h, j and k. Can you predict what angles h, j and k will be when you know angles c and d? Without working it out for every windmill pattern, how can you be certain your prediction will always work? Page 2 of 2

3 ## F is the centre of a regular pentagon. Work out the value of angle x without using an angle measurer. You MUST explain how you worked out your answer. 2 marks ## In this diagram AB is parallel to CD. Work out the value of angle x. Do not use an angle measurer. mark Page 3 of 2

4 Calculate the value of angle y. Do not use an angle measurer. mark Q4. The shape ABCD is a rectangle. BD is parallel to EF. Calculate the sizes of the angles x and y. Do not use an angle measurer (protractor). 2 marks Page 4 of 2

5 Q5. Triangle ABC is equilateral. Calculate the size of angle x. Do not use an angle measurer (protractor). x = mark Q6. Rajiv makes this circular spinner. Calculate the probability of scoring 5 on Rajiv s spinner. Give your answer as a fraction. mark Page 5 of 2

6 Vicky makes this rectangular spinner. All the sections have equal areas. She says, 'All the numbers on my spinner have the same probability of coming up'. Explain why Vicky is not correct mark Page 6 of 2

7 Q7. This is a centimetre grid. On the grid draw a triangle which has an area of 7.5cm 2 and which has an obtuse angle. Use a ruler. 2 marks Q8. Here is the start of a sequence of shapes using rectangles and triangles Each rectangle has been numbered. The pattern continues to grow in this way. Page 7 of 2

8 How many triangles will there be in the shape that has 50 rectangles in it? mark T stands for the number of triangles in each shape. R stands for the number of rectangles in each shape. What is the rule connecting T and R? mark Q9. The diagram shows two overlapping squares and a straight line. Calculate the value of angle x and the value of angle y. Page 8 of 2

9 Do not use a protractor (angle measurer). x = mark y = mark Q0. Sarah makes a pie chart to show the proportion of boys and girls in her class. Number in class Size of angle on pie chart Boys 4 44 Girls 2 26 Page 9 of 2

10 The next day another boy joins Sarah's class. She makes a new pie chart. Calculate the angle for boys on the new pie chart. 2 marks Q. The diagram shows two shaded equilateral triangles. Calculate the size of the angle x and angle y. Do not use a protractor (angle measurer). x = y = 2 marks Page 0 of 2

11 Q2. The diagram shows a right-angled triangle and three parallel lines. Calculate the size of angle x and angle y Do not use a protractor (angle measurer). x mark y mark Q3. The diagram shows a pentagon. Not drawn accurately Each side of the pentagon is the same length. Page of 2

12 Is the shape a regular pentagon? Circle Yes or No. Yes / No Explain your answer. mark Work out the size of angle a 2 marks Page 2 of 2

13 Q4. The dotted line is a diagonal of this rhombus. 3 marks Page 3 of 2

14 M. Turning triangles solutions and what to look for Solutions Part (a) a = 72, b = 54 Notes: This question is designed to be straightforward. It gives children the confidence to move on. (b) 36, 72, 72 (c) Number of sides Angle a Angle b Part 2 (a) c = 72, d = 36 Notes: Children may be encouraged to consider why two of the angles in this triangle are the same as the triangle in part (b). (b) c = 72, d = 36 h = 08, j = 54, k = 8 Notes: Children may find it useful to refer back to part (a). (c) d = 80-2c h = 80 c j = (80 c) k = j d Notes: Other ways of expressing the relationships are possible. Some children will wish to investigate specific polygons before considering how to be certain that their findings will always work. Other children will be able to use algebra to prove that the relationships must always hold. Page 4 of 2

15 Reviewing mathematical achievement Level 5 Typically children working at level 5 are able to apply their knowledge of angle properties to identify angles within the triangles. In part (c), they work within the given constraints and check their results, considering whether their solutions are sensible. They understand that identifying 2 polygons is insufficient to constitute certainty that no other such polygons exist. Their mathematical communication, both written and oral, is clear enough that others can follow their logic. In part 2 of the task, they can apply spatial reasoning to understand how the windmills are constructed, and in part (a) can work out correctly the values of angles c and d. They express simple relationships between the angles, for example h + c = 80. Level 6 Typically children working at level 6 are able, in part (c), to give a reasoned argument as to how they know there are no more than 2 polygons. They show evidence of being systematic and logical, for example by considering factors in order with explanations as to why factors are accepted or rejected. In part 2 of the task, they apply geometrical reasoning to work out the values of all the angles in the regular pentagon. They show understanding of the given diagram by reasoning generally for other windmill patterns, and they are able to establish more complex relationships between the angles, for example h = 2j, even if they need encouragement to express these relationships using formal algebra. M2. Award ONE mark for the correct answer of 08 Award ONE mark for appropriate explanation, eg: regular pentagon, angles are 08 isosceles triangles, 2 54 Up to 2 [2] M3. (a) 40 (b) 25 [2] Page 5 of 2

16 M4. Award TWO marks for the correct answers x = 25 AND y = 45. If the answers are incorrect award ONE mark for either x = 25 OR y = 45 OR the sum of x and y being 270. up to 2 [2] M5. 32 [] M6. (a) OR 0. recurring OR % Accept equivalent fractions eg Accept 0. or %. Do not accept answers in words, eg out of 9 OR in 9 OR ratios, eg :9 (b) Explanation which recognises that all the numbers are not equally likely to come up because the angles formed at the centre by each section are not equal, eg Some are narrower at the centre than others ; The angles in the centre aren t equal. Do not accept vague or arbitrary explanations such as It s just luck ; Some have more space ; They never are equal. [2] 2 M7. Award TWO marks for any obtuse-angled triangle with an area of 7.5cm, eg Page 6 of 2

17 If the answer is incorrect, award ONE mark for any triangle with an area of 7.5cm (irrespective of angles) Accept any obtuse-angled triangle with appropriate base and height each correct to within 2mm The triangle need not have vertices on the grid intersections. Accept a triangle not drawn with a ruler, provided the vertices are correctly placed. Up to 2 2 [2] M8. (a) 98 (b) T = 2R 2 OR R = Accept equivalent expressions, eg T = R 2 2 T = 2 (R ) R = + Accept answers in words, eg to get T, you times R by 2 and then you take away 2 ; it s less than R, then you double it and that s T. [2] M9. (a) 55 If answers for 9a and 9b are transposed, but otherwise correct, award the mark for 9b only (b) 25 [2] Page 7 of 2

18 M0. Award TWO marks for the correct answer of 50 If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg = Calculation need not be completed for the award of the mark. Up to 2 [2] M. (a) x = 55 (b) y = 85 If answers for 5a and 5b are transposed, but otherwise correct, award ONE mark only, in the 5b box. [2] M2. (a) x = (b) y = OR y = (Answer to (a) 35 ) If answers to x and y are transposed but otherwise correct, award ONE mark only in the (b) box. [2] M3. Indicates No and gives a correct explanation eg The angles are not the same size A regular pentagon looks like this, with its angles all the same size All the angles should be 08 It doesn t have rotation symmetry It s got more sides than a square so all its angles should be obtuse, but they re not Page 8 of 2

19 60 2 Shows that the 50 angle can be split into 90 and 60 or Divides the pentagon vertically and shows that half a is 30 or Draws triangles to show a rectangle, labelling the non-right angles on at least one side correctly eg or Shows or implies that the angle sum of a pentagon is 540 Accept minimally acceptable explanation eg Different angles A regular pentagon doesn t have right angles in it A regular one can t have 50 angles It doesn t look the same when it s turned Not all the angles are obtuse! Incorrect angle size for a regular pentagon given Condone alongside a correct response eg, accept The angles are different, they should be 60 (error, but all equal implied) The angles should all be 70 (error) eg, do not accept The 90 angles should be 60 (does not imply the angles should all be the same) Page 9 of 2

20 Do not accept incomplete explanation eg Not the same It has two right angles Two angles are the same A regular pentagon looks like this A regular pentagon doesn t have any vertical lines! Indicates Yes, or no decision made, but explanation clearly correct Condone provided the explanation is more than minimal [3] M4. b = 50 a = 20 U As evidence of a correct method, in either part, shows or implies that the angles in one of the triangles are a, b and b eg, in the first question part 80, 50, 50 seen (80 80) 2 (360 60) 2 2 eg, in the second question part ( ) 2 eg, correct answers transposed! Incomplete or no working shown Provided at least one correct angle is credited, award this mark! In the second question part 80, 80, 20 is insufficient without any indication of the position of the equal angles [3] Page 20 of 2

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