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1、Mathematical Investigation: Paper SizeThe purpose of this worksheet is to investigate the relationship between A1 paper, A2 paper, A3 paper, A4 paper, etc.1. Given a sheet of A3 paper (white) and a sheet of A4 paper (blue), try to figure out and write down any two relationships between them.2. Measu
2、re the length and width of the A3 and A4 papers and record your observations in the relevant portion of the table below. Calculate and record the ratio “length / width” for these paper sizes, correct to 4 significant figures (s.f.).PaperLength / cmWidth / cmLength / Width1.A12.A23.A34.A45.A56.A67.A7
3、8.A89.A910.A103. Given another sheet of A4 paper (green), fold it into two equal halves along the dotted line as shown below. Cut along the dotted line. Each half is called an A5 paper. Measure the length and width of the A5 paper and calculate the ratio “length / width”, correct to 4 s.f. Record yo
4、ur observations in the above table.4. Given another sheet of A5 paper (yellow), fold it into two equal halves along the dotted line as shown above. Cut along the dotted line. Each half is called an A6 paper. Repeat the same measurements, calculation and recording.5. Given another sheet of paper of A
5、6 paper (pink), cut out an A7 paper and repeat the same measurements, calculation and recording.6. Arrange the A4, A5, A6 and A7 papers on the A3 paper according to the diagram below. The biggest rectangle shown in the diagram is the A3 paper.A4A5A6A77. Using the remaining A5 paper (green), A6 paper
6、 (yellow) and A7 paper (pink), cut out an A8 paper (green), an A9 paper (yellow) and an A10 paper (pink). Repeat the same measurements, calculation and recording.8. Arrange the A8, A9 and A10 papers in the same pattern as the diagram in Q6.9. There are other ways to arrange the papers. Can you arran
7、ge them in such a way that the papers “spiral” inwards? Sketch the arrangement below.10. Observe the ratio “length / width” in the above table. What do you notice?11. Can you recognise what the value of this ratio is? If yes, what is it? If not, just answer no.12. We will find the exact value of thi
8、s ratio from another approach. The bigger rectangle shown in the diagram is A4 and each half is A5.A5A5Let the length and width of the A4 paper be x and y respectively. Let the ratio “length / width” be r. Express the length, x, of the A4 paper in terms of y and r.13. Express the length of the A5 pa
9、per in terms of x, y and/or r.14. Express the width of the A5 paper in terms of its length and r.15. Express the width of the A5 paper in terms of the length of the A4 paper.16. Using the expressions in Q12, Q14 and Q15 to form an equation and solve for r.17. Is the value of r obtained in Q16 the sa
10、me as the ratio “length / width” obtained from the table?18. From the table, you can observe that the length of an A4 paper is twice the width of an A5 paper, and the length of an A5 paper is twice the width of an A6 paper, etc. Can you explain why this is so? Hint: Use the exact value of the ratio
11、r.19. Without the use of calculator, complete the rest of the table for the length and width of an A1 and an A2 paper. You can use the calculator to evaluate the ratio “length / width” and compare it with the exact value of r.20. In order to save paper when photocopying, sometimes we may like to pho
12、tocopy two A4 pages to a single A4 page:A4A4A4What is the percentage of reduction that you need to set in the photocopy machine? Note that the percentage of reduction refers to the reduction of the length or width, not the area.21. At other occasions, we may like to enlarge an A4 size to a big poste
13、r of A3 size. What is the percentage of enlargement that you need to set in the photocopy machine?22. One of the most important questions in any mathematical investigations that you should ask yourself is: “What else is there for me to investigate?” List one thing that you would like to investigate
14、further and if time permits, investigate it.Topic:Sec 1: Numbers (Square Roots). But Q12-16 involve solving simple algebraic equations (Sec 1).Materials needed for Each Group of 4 Students:ï One pair of scissorsï One sheet of A3 paper (white)ï One sheet of A4 paper (blue)ï One sh
15、eet of A4 paper (green)ï One sheet of A5 paper (yellow): take an A4 paper and divide into 2 equal halves as shown in Q3ï One sheet of A6 paper (pink): take an A4 paper and divide into 4 equal quarters similar to Q3ï Magnets to paste papers on whiteboard (for teacher)ï Pupils must
16、 bring their own ruler and calculatorAnswers:1.A3 paper is twice the size of A4 paper; width of A3 paper = length of A4 paper2-5, 7, 19.The measurements below are standard sizes. However actual sizes may be slightly different due to manufacturers faults.PaperLength / cmWidth / cmLength / Width1.A184
17、59.41.4142.A259.4421.4143.A34229.71.4144.A429.7211.4145.A52114.81.4196.A614.810.51.4107.A710.57.41.4198.A87.45.251.4109.A95.253.71.41910.A103.72.61.4239.A5A4A9A8A6A710.About the same. Differences in actual length and width are due to manufacturers faults. Round-off errors in the length and width can
18、 also cause the ratio to be different.11.It is ok if the students cant recognise this number because we will calculate this number from another approach.12.Length of A4 paper, x = ry13.Length of A5 paper = y (width of A4 paper)14.Width of A5 paper = y / r (length of A5 paper / r)15.Width of A5 paper
19、 = x / 2 (length of A4 paper / 2)or = ry / 2 from Q1216.y / r = x / 2 Þ y / r = ry / 2 (from Q12) Þ r 2 = 2 Þ r = Ö2 » 1.414217.Yes (or about the same due to problems stated in answer to Q10)18.length of A4 paper= Ö2 ´ width of A4 paper= Ö2 ´ length of A5
20、 paper= Ö2 ´ Ö2 ´ width of A5 paper= 2 ´ width of A4 paper20.width of A4 paper / width of A3 paper = width of A4 paper / length of A4 paper= 1 / Ö2= 0.707 (to 3 s.f.)or round down to 70% (because 71% is still too big)21.Ö2 = 1.41 (to 3 s.f.) or 141%22.Examples:
21、9; What is the smallest A-series paper manufactured? Ans: I dont knowï How do you visualise how big an A1 paper is? Ans: 4 times the size of an A3 paperï How do you construct a rectangle that has the same shape as an A4 paper using only straightedge and compasses? Ans: See below or Arithmetic_Rectangle.gsp. The gist of it is: ABCD is a square with AB = 1 unit; AC is a diagonal so that ÐDAE = 45° DAEF is a right-angled isosceles D so that AF = Ö2 units.ABCDEF