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
1.
A1
2.
A2
3.
A3
4.
A4
5.
A5
6.
A6
7.
A7
8.
A8
9.
A9
10.
A10
Length / cm
Width / cm
Length / Width
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.
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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.
A5
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
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a where a is a whole number.
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.
A5
A5
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.
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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.
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Mathematical Investigation: Paper Size
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.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Paper
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
Length / cm
84
59.4
42
29.7
21
14.8
10.5
7.4
5.25
3.7
Width / cm
59.4
42
29.7
21
14.8
10.5
7.4
5.25
3.7
2.6
Length / Width
1.414
1.414
1.414
1.414
1.419
1.410
1.419
1.410
1.419
1.423
9.
A5
A4
A8
A9
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.
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11.
It is ok if the students can’t recognise this number because we will calculate this number from
another approach.
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.
F
D
C
E
A
B
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
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