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Contextualised task 39

Fun with Flags

Teaching notes This task focuses on the mathematical specification of flags. Students will first consider the flag of Wales, and then of the UK. They will see that there are very particular ways of defining a flag using ratios, and consider how to represent these facts using fractions too.

Task 1: From fractions to ratios Outline Students begin to explore the mathematical specifications to the way in which flags are constructed. They investigate the specifications for flags of Nordic countries, moving between fractions to ratios to describe their construction. You will need: Teachers’ script PowerPoint Question sheet Flags of Nordic countries sheet Squared paper Mark scheme

Task 2: From ratios to fractions Outline Students now consider the information that was first provided as ratios. They use their understanding developed in the first part of the task to rewrite these statements using fractions. Students are asked to write paragraphs, and they may want to use a process of drafting and redrafting, as they would do in English. You will need: Question sheet Squared paper Mark scheme

Task: Teachers’ script for PowerPoint presentation

The text in the right-hand boxes provides a possible script to be read to students. However, it is probably preferable to use your own words and elaboration. When questions are asked, time for discussion in pairs / groups should be provided. Ensure that students are given opportunity to explain their reasoning in response to these questions. All students need to understand the concepts in order to make progress with the task. Slide 1

Flags

Keep this slide on the screen until you are ready to start the presentation

Slide 2

All flags are constructed using precise mathematical information. A correct Welsh flag should have its sides in the ratio 10:6. Tell me another ratio equivalent to 10:6.

(e.g. 5:3, 50:30)

If a Welsh flag is one metre wide, how tall should it be?

(60 cm)

The white and green background comprises equally sized areas.

The dragon is the only part of the Welsh flag that is not standardised. But it should be the correct size and in the right place. What does this diagram tell you about the size and position of the dragon?

(e.g. For every 10 units across on the flag, the dragon is 8 units across. The box in which the dragon is drawn must be placed centrally on the flag.)

Slide 3

Here is a diagram of the flag of England. Work with a partner to find 8 mathematical facts about this flag.

You could give hints about ratio, area, enlargement

For example:

If the flag is 50 cm wide it must be 30 cm tall The ratio of length to width is 5:3 (although 3:5 would

also be correct as length does not necessarily imply the longest side)

If the flag is 1 metre wide, the area of white is 4 × 24 × 44 = 4224 cm2.

The ratio of division along the top edge of the flag is 11:3:11.

Ensure that examples such as those on the above list are discussed

Slide 4 Slide 5

25 256

2

6

10

2

2

6

6

10

2

The Union Flag (sometimes referred to as the Union Jack) is an attempt to combine the flags of countries in the United Kingdom (The Union).

Advance one click

Here are the flags of England, Scotland and Northern Ireland. At the time of the creation of the flag, (and the union of kingdoms) over 200 years ago, Wales was part of the ‘Kingdom of England’. Therefore, its own flag was not integrated.

This is a complex flag to construct. If, first, the diagonal lines are ignored, the horizontal and vertical lines can be drawn.

Advance two clicks

Then lines can be drawn that join opposite corners.

Advance 2 clicks

These diagonal lines can then be used to find all remaining boundary lines.

Advance three clicks

Slide 6

This version of the union flag has an aspect ratio of 1:2. It is twice as long as it is wide.

Most countries have a variant of their flag for particular purposes. The UK flag can also be constructed using an aspect ratio of 3:5.

Advance one click

This is the version of the union flag used by the army. This ‘war flag’ is sometimes referred to as an ensign instead of a flag.

Fun with Flags Task A: From fractions to ratios Throughout this task the word ‘length’ will refer to the longest side of a rectangle, and ‘width’ to the shortest side. The ‘aspect ratio’ of a flag is the ratio of width to length in its simplest form. Not all the flags you will see here are rectangular. If this is the case then the aspect ratio is the ratio of maximum width to maximum length.

This is the state flag of Sweden. It has an aspect ratio of 5:8. Horizontally the colours are split in the ratio 5:2:9. Vertically, the colours are in the ratio 4:2:4.

The Swedish naval ensign has an aspect ratio of 1:2. It also has the colours in the ratio 4:2:4 vertically. At the short end of the ‘tails’, the colours are split in the ratio 5:2:5. At the long end of the tails the ratio is 5:2:13. Overall, we can say that the colour split is 5:2:5:8. Check that you can see why.

The table on Task A: Flags of Nordic countries shows information about two flags from each of Norway, Sweden, Finland, Denmark and Iceland. 1. Find each missing aspect ratio in the table. 2. The Danish and Icelandic flags have information in the notes column. Use this information

to find each colour split as a ratio. You need to state both horizontal and vertical colour splits in each case, as in the examples above.

Note The Icelandic state flag could have a horizontal colour split defined: at the end of the diagonal cut (as with the Norwegian state flag given) at the corner of the blue trapezium, or at both the end of the diagonal cut and the corner of the trapezium State which your solution refers to.

Task A: Flags of Nordic countries

Flag Image Aspect

ratio Notes

Sweden: state ensign

5:8 The colours are split 5:2:9 horizontally and 4:2:4 vertically

Finland: civil flag

The colours are split 4:3:4 vertically and 5:3:10 horizontally

Denmark: civil flag

14:17 “The white cross must be 1/7 of the flag's height. The two first fields must be square in form and the two outer fields must be 6/4 lengths of those.” From the Koffardiflaget. Source: https://en.wikipedia.org/wiki/Flag_of_Denmark

Norway: civil flag

The colours are split 6:1:2:1:6 vertically and 6:1:2:1:12 horizontally

Iceland: civil flag

18:25 “The arms of the cross extend to the edge of the flag, and their combined width is 2/9, but the red cross 1/9 of the combined width of the flag. All blue areas are rectangles. The smaller blue areas are square and the outer blue areas as wide as them, but twice the length.” Adapted from the Law of the National Flag of Icelanders and the State Arms. Source: https://en.wikipedia.org/wiki/Flag_of_Iceland

Sweden: naval ensign

1:2 The colours are split 4:2:4 vertically and 5:2:5:8 horizontally.

Finland: naval ensign

The colours are split 4:3:4 vertically and 5:3:6:5 horizontally.

Denmark: state flag

56:107 "The cross must be 1/7 of the flag's height. The two first fields must be square in form with the height of 3/7 of the flag's height. The two outer fields are rectangular and 5/4 the length of the square fields. The tails are 6/4 the length of the rectangular fields." From the Kongeflaget. Source: https://en.wikipedia.org/wiki/Flag_of_Denmark

Norway: state flag

The colours are split 6:1:2:1:6 vertically and 6:1:2:1:6:11 horizontally.

Iceland: state flag

9:16 “The state flag differs from the civil one, that the larger blue rectangles are three times longer than the smaller blue rectangles and split at the end, each cut directly from the outer corners through their centre line. This cuts the inner edge of each larger rectangle at 4/7 of outer length and 3/7 of inner length. When this cut encounters the edge of the red cross it is cut vertically.” Adapted from the Law of the National Flag of Icelanders and the State Arms. Source: https://en.wikipedia.org/wiki/Flag_of_Iceland

Task A: Mark scheme

The information below is intended as a guide only Full credit Finds the ratios as stated in the table below: Flag Image Aspect

ratio Notes

Sweden: state ensign

5:8 The colours are split 5:2:9 horizontally and 4:2:4 vertically

Finland: civil flag

11:18 The colours are split 4:3:4 vertically and 5:3:10 horizontally

Denmark: civil flag

14:17 The colours are split 3:1:3 vertically and 6:2:9 horizontally

Norway: civil flag

8:11 The colours are split 6:1:2:1:6 vertically and 6:1:2:1:12 horizontally

Iceland: civil flag

18:25 The colours are split 7:1:2:1:7 vertically and 7:1:2:1:14 horizontally

Sweden: naval ensign

1:2 The colours are split 4:2:4 vertically and 5:2:5:8 horizontally.

Finland:

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