What is open? (And what is closed?)

1. What is open?(And what is closed?)

2. The answer is ‘5’ - What is the question?

3. Read this: “ ‘Twas brillig, and the slithy toves Did gyre and gimbal in the wabe; All mimsy were the borogoves, And the mome raths outgabe.” (from Jabberwocky by Lewis Carroll) Now answer these questions: 1. What were the slithy toves doing in the wabe? 2. How would you describe the state of the borogroves? 3. What can you say about the mome raths?

4. Read this: “ ‘Twas brillig, and the slithy toves Did gyre and gimbal in the wabe; All mimsy were the borogoves, And the mome raths outgabe. • And now answer these questions: • 4. Did you need to understand the text in order to answer questions 1 to 3? • 5. Why were the borogroves mimsy? • 6. How effective was the mome raths’ strategy?

5. “The activities in this resource encourage a different approach to learning. They are designed to promote what Marton and Saljo* called ‘deep learning’. The characteristics of this include a desire to understand and extract meaning. Adopting this approach is likely to lead to better exam results and longer-term recall. Learners tend to adopt the ‘surface’ approach if they believe that this is what the assessment process requires.” *..and West-Burnham - Engineering Resources pd session 6 ‘questioning for learning’

6. Bloom/Anderson and Kratwohland all that • http://www.learningandteaching.info/learning/bloomtax.htm

7. Whole class questioning • Looked for by lesson observers • Include in preparation • Reflect on the proposed question • Pursue each open question for understanding – one excellent question might be enough!

8. Thinking time • Some learners want longer to think about their answer, and others give an answer straight away. We all know that it is important not to rush the thinking process. What strategies do you use to solve this problem?

9. Death by textbook – a cure is at hand!

10. Find the mean, mode, median and range of the numbers: 3, 7, 9, 1, 3, 14, 7, 3, 4, 9 1. Answer the question. 2. If I change a 3 for a 4 what happens to the measures? 3. Can you change one number so that the mean changes but none of the other measures change? 4. Can you change one number so that the median changes but none of the other measures change? 5. If an extra 3 is added what will change? 6. Can you change one number so that the range is reduced but none of the other measures change? 7. How can you change just one number so that two of the measures change?

11. Question: The monthly charge for a mobile phone is £25. This includes 300 minutes free each week. After that there is a charge of 5p per minute. Calculate the cost of using the phone for 540 minutes, 600 minutes, 310 minutes and 450 minutes for each of the 4 weeks in one month. 1. What other questions could you ask about this situation? 2. What would happen to the bill if only 100 minutes were used in the third week? 3. What difference is it likely to make to the bill if calls after 6.00 pm are only 3p per minute? 4. Is it good value? 5. What other information would you want to know about the charges for this phone before you agreed to have it? 6. Can you make the question more realistic?

12. Question: Tim buys a pack of 12 cans of cola for £4.80. He sells the cans for 50p each. He sells all of the cans. Work out his percentage profit. 1. Change some or all of the numbers so that the percentage profit is still the same. 2. If he can get 24 cans for £9.20 should he buy them? 3. He wants to make at least 10% profit. What should he do? 4. The price has gone up to £5.20 for 12 cans, what should he do to maintain his percentage profit. 5. What else could you ask about this situation?

13. Question: In a sequence u1 = 3, u2 = 9 and u3 = 15. Find u6 and a formula for un. 1. Is 90 a term in the sequence? 2. Are all terms a multiple of 3? How do you know? 3. Can you find other sequences with the same u3 and u6? Describe their patterns. 4. How could you change the question so that 100 is a term of the sequence? 5. What would you have to change to get u6 = 40? 6. What difference would it make if u1 = 4? 7. What could the next part of the question be?

14. Prompts to open up closed questions: 1. What happens when …..? 2. What happens if I change …..? 3. What difference would it make if …..? 4. What would the next part of the question be? 5. What happens next ….? 6. How could you make it true that …..? 7. What could you change so that …..? 8. How could you make the question easier …..? 9. How could you make the question harder …..?

15. A young person’s railcard gives one third off the normal price. Jenny uses her railcard to buy a ticket. The normal price of the ticket is £36.45. Work out how much she pays for the ticket. 2. A text book costs £12.99. Work out the cost of 20 books. A teacher can afford to buy 12 of the books. Write 12 out of 20 as a percentage. 3. Andy spends one third of his pocket money on a computer game and one quarter on a ticket to a football match. Work out the fraction of his pocket money that he had left.

16. 4. Bronze is made from copper and tin. The ratio of the weight of copper to the weight of tin is 3 : 1. (a) What weight of bronze contains 36 grams of copper? (b) Work out the weight of copper and the weight of tin in 120 grams of bronze. 5. (a) Change 10 kilograms to pounds. (b) Change 7 pints to litres. 6. Express y = x2 + 8x + 3 in the form (x+a)2+b where a and b are integers to be determined. Hence write down the transformation that sends y= x2 to the graph Y = x2 + 8x + 3.

17. 8cm 2cm 4cm 10cm 7. The diagram shows 3 small rectangles inside a large rectangle. The large rectangle is 10cm by 8 cm. Each of the small rectangles is 4cm by 2cm. Work out the area of the region shaded in the diagram.

18. A baker offers a 3 tiered wedding cake. Each layer is cylindrical. The bottom layer has diameter 29cm, the middle layer has diameter 23cm and the top layer has diameter 17cm. What other mathematics could be linked to this question? Write questions that use as many different aspects of mathematics as you can think of. What extra information would you need in order to answer your questions?

19. Percentage of male and female who had drunk alcohol on 5 days or more in the week prior to interview. Mean consumption of children aged 11 – 15 who drank in the last week. What questions could you ask? Can you write a hard question, a medium question and an easy question all of which can be answered from the graph or the chart above? What other questions could you ask?

20. 10 cm 30º What questions could you ask?

21. x (3, 10) x (1, 4) What questions could you ask?

22. What questions could you ask?

23. Distance from home Time What questions could you ask? What information would you need in order to answer each question?

24. Trapezoidal tables are put together in pairs so that they will seat 6 students. What questions could you ask? What information would you need in order to answer each question?

25. Questioning for Pythagoras’ Theorem

26. Type 1 Type 2 x cm x cm 8cm 13cm 6cm 7cm Type 3 Type 4 13cm 11cm 12cm 8cm x cm x cm

27. y 15 x Type 5 Type 6 18 Find the length of the diagonal of the rectangle. 8cm 24 15cm Find x and y.

28. 8 cm and 10cm are the lengths of two sides of a right angled triangle. Find possible triangles and use Pythagoras to justify that they are indeed right angles triangles. The hypotenuse of a right angled triangle is 10cm. Find possible triangles and use Pythagoras to justify that they are indeed right angles triangles. One of the sides of a right angled triangle that is adjacent to the right angle is 10cm. Find possible triangles and use Pythagoras to justify that they are indeed right angles triangles. One side of a right angled triangle is 10cm. Find possible triangles and use Pythagoras to justify that they are indeed right angles triangles.

29. Is this diagram possible? Justify your answer. 15 y 15 39 Find x and y. x 18 18 24 24 Not to scale. Put possible lengths on the diagram so that the right angles work. Justify your decisions. Can you generalise? Instead of : How about: Or this one:

30. Creating more thinking problems • All problems should make you think but the thinking problems • are problems that are being focussed on here: • make you stop and think and keep you thinking. • explore understanding. • are usually open with more than one answer. • are easily accessible but can be extended very naturally into something quite complex. • require some problem solving and reasoning skills. • cannot be solved just by trial and error or rote learned rules

31. Original Problem: Plot the points A (1, 1), B (2, 3), C (4, 4) and D (3, 2). Join them up to create the shape ABCD and name this shape. Thinking Problem: The coordinates of one vertex of a rhombus are (3, 4). Give possible coordinates for the other vertices.

32. Original Problem: Plot the points A (1, 1), B (2, 3), C (4, 4) and D (3, 2). Join them up to create the shape ABCD and name this shape. Thinking Problem: The coordinates of one vertex of a rhombus are (3, 4). Give possible coordinates for the other vertices and justify that your shape is a rhombus in as many different ways as possible.

33. 8 cm 9cm Area of the sloping face is 48cm² 6cm Original Problem: Find the volume of this triangular prism. State the units of your answer. Thinking Problem: Find possible dimensions for the prism. Which give the maximum volume? Does this prism have the largest surface area?

34. 6 3 2 8 5 4 10 5 6 Original Problem: Find the area of each shape.

35. x 3 2 3x 5 x 10 5 x Thinking Problem: Find the area of each shape in as many different ways as you can and show that each gives the same answer.

36. Original Problem: Construct two box plots from the stem and leaf: 3 4 8 7 3 3 1 3 5 5 7 3 6 8 6 2 1 1 0 3 4 4 5 7 5 5 5 2 1 6 8 8 8 9 5 7 7 9 6 3 1 4/5 represents 4.5 cm

37. Thinking Problem: Construct a possible back to back stem and leaf that is represented by these box plots.

38. Original Problem: A driver covers 110 km in 3 hours and then travels at 70 km h-1 for a further 2 hours. What is his average speed? Thinking Problem: The average speed of a journey is 60 km h-1. Construct a possible travel graph. Justify that its average speed is 60 km h-1 and describe the journey.

39. x (2, 6) x (6, 3) Find a possible equation of the line. Justify your answer. Can you generalise your answer?

40. 2y + 3 x = 5 Find a possible equation for the other line. Justify your answer. Can you generalise your answer?

41. Give a possible equation for the circle. Justify your answer. y = x