Staged problem solving. Blue card sort. Areas of Circles: Staged Problem solving. Card sort, structured stages, independence

Transcription

1 Areas of Circles: Staged Problem solving Card sort, structured stages, independence 1. The diagram shows a semicircle drawn inside a rectangle. Blue card sort The semicircle has a diameter of 8 cm. The rectangle is 8 cm by 4 cm. Work out the area of the shaded region. Give your answer correct to 3 significant figures. [5 marks] Page 1 of 9

2 2. Green card sort The arc ABC is a quarter of a circle with centre O and radius 4.8 cm. AC is a chord of the circle. Work out the area of the shaded segment. Give your answer correct to 3 significant figures. [5 marks] Page 2 of 9

3 3. The diagram shows a trapezium ABCD and two identical semicircles. The centre of each semicircle is on DC. Work out the area of the shaded region. Give your answer correct to 3 significant figures. Write down the formula for the area of the trapezium Substitute a = 28, b= 12 and h= 14 into formula Find calculator answer for area Write down formula for area of a circle Substitute D= 6 (why D=6?) and find the area of a circle Use area of circle to find area of two semicircles Find calculator answer for shaded area (trapezium 2 x semi-circles) Give answer to required degree of accuracy [6 marks] Page 3 of 9

4 4. Here is a diagram showing a rectangle, ABCD, and a circle. Staged problem solving BC is a diameter of the circle. Calculate the percentage of the area of the rectangle that is shaded. Give your answer correct to 1 decimal place. Work out the area of the rectangle Write down the formula for the area of a circle Substitute r = 8 into formula (why r= 8?) Find area of semi-circle Find shaded area (area of rectangle area of semi-circle) Write shaded area as a proportion of full rectangle area shaded Area of rectangle Multiply this proportion by 100 to make it a % Give answer to required degree of accuracy [6 marks] Page 4 of 9

5 5. This shape is made from circles. The diameter of the smaller circle is 10 cm The diameter of the larger circle is 20 cm What fraction of the shape is shaded? Give your answer in its simplest form. [4 marks] Page 5 of 9

6 Additional questions 6. The diagram shows a circle inside a semicircle. The circle has a diameter of 10 cm. The semicircle has a diameter of 20 cm. Work out the area shaded. Give your answer correct to 1 decimal place. [5 marks] 7. The diagram shows a circular pond with a path around it. 5 m 1 m The pond has a radius of 5m. The path has a width of 1m. Work out the area of the path. Give your answer correct to 3 significant figures. [3 marks] Page 6 of 9

7 8. The diagram shows a rectangle inside a semicircle. The rectangle has dimensions 16 cm by 6 cm Work out the shaded area. Give your answer in terms of π. *This question requires the use of Pythagoras to find the radius of the circle. [4 marks] 9. Four identical circles just fit inside a square as shown. 12 cm Work out the area of the shaded section. Give your answer in terms of π. [4 marks] 10. The diagram shows a semicircle of diameter 14 cm inside a rectangle. 12 cm Work out the shaded area. 14 cm [4 marks] Page 7 of 9

8 Teaching notes The resources for each topic include: 1. A set of problems (word document) a. First two problems are supported with a card sort which includes a description about what you are doing and the calculation necessary for each stage. b. Two further problems with a tabulated structure including the description of each stage (no calculation or formula given). c. Unstructured question(s) for pupils to systematically solve. Additional questions are not included on the ppt and could be printed separately (to reuse perhaps). 2. Cards sorts (word document) Colour coded so that the students can distinguish between problem 1 and 2 solution but the teacher can print on single pages. Alternatively, this document can be adapted so that all of one problem is printed on coloured paper (this will cost less in printer ink). 3. Powerpoint For question 1 and 2 the correct order for the card sort is given followed by the mark allocation. For question 3 and 4 solutions are added step by step followed by the mark allocation. For Question 5 a table describing each step, followed by each stage of the working and finally the mark allocation. Card sort, Structured stages, independence The structure of the materials is to enable learners to access problems and encourage them to systematically record solutions before they are asked to complete unstructured questions independently. The card sort includes both the working and a description for each stage of the solution. Students are asked to order the solution in a logical step by step manner. The second phase give students a table describing what they should be doing but does not include formulas or working. The final phase gives students unstructured problems for them to systematically record their own solution. Answers: cm cm cm % cm m Page 8 of 9

9 Find area of rectangle 8 x 4 = 32 cm² Write down formula for area of circle A = πr² Substitute r = 4 into formula (why r = 4?) A = π x 4² Find calculator answer (full display) A = cm² Divide area by 2 for semi-circle Shaded area is = Give answer to required degree of accuracy 6.87cm² Write down the formula for the area of a circle A = πr² Substitute r = 4.8 into formula A = π x 4.8² Find calculator answer (full display) A = cm² Divide area by 4 for quadrant Area of quadrant (i.e. 4) = cm² Find area of triangle (OC = OA = radius) A = (4.8 x 4.8 ) 2 = 11.52cm² Shaded area is Quadrant - triangle Shaded area = = cm² Give answer to required degree of accuracy Shaded area = 6.58cm² Page 9 of 9