Mathematical Investigation: Paper Size

The 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. Measure 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.).

Paper Length / cm Width / cm Length / Width 1. A1 2. A2 3. A3 4. A4 5. A5 6. A6 7. A7 8. A8 9. A9 10. A10

3. 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 your 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 A6 paper (pink), cut out an A7 paper and repeat the same measurements, calculation and recording.

A4 A7 A6

7. Using the remaining A5 paper (green), A6 paper (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 arrange 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. Express the ratio “length / width” in the form a where a is a whole number.

© Joseph Yeo 2 12. We will look at the problem from another angle. The bigger rectangle shown in the diagram is A4 and each half is A5. The question is why is the length of the A4 paper twice the width of the A5 paper.

Since the length of the A4 paper is 2 times the width of the A4 paper, and the length of the A5 paper is also 2 times the width of the A5 paper, find the length of the A4 paper in terms of the width of the A5 paper.

Length of A4 Paper = 2  Width of A4 Paper

13. 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.

14. In order to save paper when photocopying, sometimes we may like to photocopy two A4 pages to a single A4 page:

A4 A4 A4

What 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.

15. At other occasions, we may like to enlarge an A4 size to a big poster of A3 size. What is the percentage of enlargement that you need to set in the photocopy machine?

P.T.O.

© Joseph Yeo 3 16. 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 further and investigate it.

Question: ______

Solution:

Conclusion

Write down one main lesson that you have learnt from this worksheet.

Topic:

Sec 1: Numbers (Square Roots). But Q12-16 involve solving simple algebraic equations (Sec 1).

Materials needed for Each Group of 4 Students:

 One pair of scissors  One sheet of A3 paper (white)  One sheet of A4 paper (blue)  One sheet of A4 paper (green)  One sheet of A5 paper (yellow): take an A4 paper and divide into 2 equal halves as shown in Q3  One sheet of A6 paper (pink): take an A4 paper and divide into 4 equal quarters similar to Q3  Magnets to paste papers on whiteboard (for teacher)  Pupils must bring their own ruler and calculator

Answers:

1. A3 paper is twice the size of A4 paper; width of A3 paper = length of A4 paper

2-5, 7, 13. The measurements below are standard sizes. However actual sizes may be slightly different due to manufacturer’s faults.

Paper Length / cm Width / cm Length / Width 1. A1 84 59.4 1.414 2. A2 59.4 42 1.414 3. A3 42 29.7 1.414 4. A4 29.7 21 1.414 5. A5 21 14.8 1.419 6. A6 14.8 10.5 1.410 7. A7 10.5 7.4 1.419 8. A8 7.4 5.25 1.410 9. A9 5.25 3.7 1.419 10. A10 3.7 2.6 1.423

A4 A9 A8 A6 A7

10. About the same. Differences in actual length and width are due to manufacturer’s faults. Round-off errors in the length and width can also cause the ratio to be different.

12. length of A4 paper = 2  width of A4 paper = 2  length of A5 paper = 2  2  width of A5 paper = 2  width of A4 paper

14. width of A4 paper / width of A3 paper = width of A4 paper / length of A4 paper = 1 / 2 = 0.707 (to 3 s.f.) or round down to 70% (because 71% is still too big)

15. 2 = 1.41 (to 3 s.f.) or 141%

16. Examples:  What is the smallest A-series paper manufactured? Ans: I don’t know  How do you visualise how big an A1 paper is? Ans: 4 times the size of an A3 paper  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 DAE = 45; AEF is a right-angled isosceles  so that AF = 2 units.

Conclusion

Some suggestions for main lesson learnt from this worksheet:  Length of A4 paper is 2 times its width  A3 paper is twice the size of A4 paper