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Home , Aspect ratio, Civil flag, Flag, Flag of Sweden, Line (heraldry), List of flags of Ireland

Contextualised task 39 Fun with

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

You will need:  Teachers’ script  PowerPoint  Question sheet  Flags of 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 to make progress with the task.

Slide 1 Keep this slide on the screen until you are ready to start Flags the presentation

Slide 2 All flags are constructed using precise mathematical information. A correct Welsh flag should have its sides in the 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 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 . 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 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 The Union Flag (sometimes referred to as the Union ) is an attempt to combine the flags of countries in the (The Union).

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

This is a complex flag to construct. If, first, the diagonal

Slide 5 6 lines are ignored, the horizontal and vertical lines can be

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2 drawn. 6

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10 Advance two clicks 6 2 2 25 6 25 Then lines can be drawn that join opposite corners.

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

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Slide 6 This version of the union flag has an 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.

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This is the version of the union flag used by the army. This ‘’ is sometimes referred to as an 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 of . 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 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. that you can see why.

The table on Task A: Flags of Nordic countries shows information about two flags from each of , Sweden, , and .

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,  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 Notes ratio Sweden: state 5:8 The colours are split 5:2:9 horizontally and 4:2:4 ensign vertically

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

Denmark: civil 14:17 “The white cross must be 1/7 of the flag's height. The flag 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 The colours are split 6:1:2:1:6 vertically and flag 6:1:2:1:12 horizontally

Iceland: civil 18:25 “The arms of the cross extend to the edge of the flag, 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 of Icelanders and the State Arms. Source: https://en.wikipedia.org/wiki/Flag_of_Iceland Sweden: naval 1:2 The colours are split 4:2:4 vertically and 5:2:5:8 ensign horizontally.

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

Denmark: state 56:107 "The cross must be 1/7 of the flag's height. The two flag 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 The colours are split 6:1:2:1:6 vertically and flag 6:1:2:1:6:11 horizontally.

Iceland: state 9:16 “The state flag differs from the civil one, that the larger flag 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 Notes ratio Sweden: state 5:8 The colours are split 5:2:9 horizontally and 4:2:4 ensign vertically

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

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

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

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

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

Finland: naval 11:19 The colours are split 4:3:4 vertically and 5:3:6:5 ensign horizontally.

Denmark: state 56:107 The colours are split 3:1:3 vertically and 24:8:75 flag horizontally.

Norway: state 16:27 The colours are split 6:1:2:1:6 vertically and flag 6:1:2:1:6:11 horizontally.

Iceland: state 9:16 The colours are split 6:1:2:1:6 vertically and flag 7:1:2:1:9:12 horizontally (end of diagonal cut) 14:2:4:2:21:21 horizontally (corner of blue trapezium) 14:2:4:2:18:3:21 horizontally (combined)

 Note that students might find constructing the flags on squared paper helpful, especially for the Icelandic flags.  Note that in the case of the Danish state flag, students could carry out some involved calculations with fractions, or they might work backwards from the aspect ratio given.

Partial credit

Finds all the aspect ratios

AND

Finds at least three of the colour splits as a ratio

Note the following calculations for the Danish state flag:

Limited credit

Finds at least three of the missing aspect ratios

AND

Finds at least two of the colour splits as a ratio

No credit

Any other response. Fun with Flags

Task B: Question

The information about the Danish and Icelandic flags stated the official state guidelines and laws. Information about the flags of Sweden, Finland and Norway was stated as a ratio. This is shown again in the table at the bottom of the page.

In the first part of this task you had to interpret the paragraphs that used fractions. For example, for the Danish state flag;

“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.”

You were then able to write the ratio of the colour split. The second part of the task reverses this process. You will need to write ratios as fractions.

The problem Write a paragraph to describe how to construct each of these six flags. The descriptions must use fractions. Your paragraphs are not allowed to contain ratios.

Flag Image Aspect Notes ratio Sweden: state 5:8 The colours are split 5:2:9 horizontally and 4:2:4 ensign vertically

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

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

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

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

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

Task B: Mark scheme

The information below is intended as a guide only

Full credit

Writes a paragraph to describe each flag. Each paragraph using fractions and/or multipliers only, and would enable an exact replica to be created.

Note that the following examples are not the only solution. Assessing this for a large number of students could be time-consuming. Peer-assessment could be utilised effectively in this case, with students challenged to use others’ instructions to construct a flag.

Note also that some students may find using squared paper helpful.

Swedish state ensign An arm of the cross has width 1/5 of the width of the flag. The two small blue rectangles are equal in size, and have a length 5/4 of their width. The two larger blue rectangles have a length 9/16 of the length of the flag.

Finnish An arm of the blue cross has width 3/11 of the width of the flag. The two small white rectangles are equal in size, and have a length 5/4 of their width. The two larger white rectangles are twice the length of the smaller ones.

Norwegian civil flag An arm of the blue-and-white cross has a width of 1/4 of the width of the flag, and the blue cross 1/8. There are two red squares, and the red rectangles have a width that is ½ of their length.

Swedish naval ensign An arm of the yellow cross has width 1/5 of the width of the flag. The two small blue rectangles are equal in size, and have a length 5/4 of their width. The shorter of the parallel sides in each trapezium is equal to the length of a blue square. The longer of the parallel sides in each trapezium is 13/20 of the length of the flag.

Finnish naval ensign An arm of the blue cross has width 3/11 of the width of the flag. The two small white rectangles are equal in size, and have a length 5/4 of their width. The shorter of the parallel sides in each trapezium is 3/2 of its height. The longer of the parallel sides in each trapezium is 5/19 of the length of the flag.

Norwegian civil flag An arm of the blue-and-white cross has a width of 1/4 of the width of the flag, and the blue cross 1/8. There are two red squares. The shorter of the parallel sides in each trapezium is equal to the length of a red square. The longer of the parallel sides in each trapezium is 11/27 of the length of the flag.

Partial credit

Completes a correct paragraph for four or five of the flags

Limited credit

Completes a correct paragraph for two or three of the flags

No credit

Any other response.

Progression in reasoning Identify processes and connections  transfer mathematical Apply skills within familiar Identify, perhaps with some Identify independently the skills across the contexts guidance, the skills needed skills needed within curriculum in a variety e.g. understands the meaning within increasingly complex increasingly complex and of contexts and of an aspect ratio and unfamiliar contexts unfamiliar contexts everyday situations e.g. converts the ratios into e.g. converts the ratios into fractions (task 2) with some fractions (task 2) without guidance guidance

Represent and communicate  explain results and Explanations are clear – both A wider range of appropriate Orally and in writing: use procedures precisely orally and in writing, using mathematical vocabulary is mathematical vocabulary using appropriate some mathematical used in explanations. precisely mathematical language vocabulary. If written as Arguments are supported e.g. uses fractions fluently to instructions, they will lead to with evidence. describe at least four of the the intended correct result. e.g. writes a paragraph to flags in task 2 e.g. writes a paragraph to describe the Swedish naval describe construction of the ensign Swedish state ensign

Review  select and apply e.g. draws a rectangular flag e.g. draws any flag on squared e.g. converts from fractions to appropriate checking on squared paper paper and labels distances to ratios, and then checks by strategies check against the ratios converting ratios back into fractions

GCSE Content GCSE Mathematics – Numeracy and GCSE Mathematics GCSE Mathematics only Understanding number and place value  Using the equivalences between fractions and ratios  Converting numbers from one form into another Understanding number relationships and methods of calculation  Addition, subtraction, multiplication and division of fractions  Expressing one number as a or of another  Calculating using ratios in a variety of situations; proportional division. Understanding and using properties of position, movement and transformation  Interpretation and construction of drawings

Key Foundation tier content is in standard text. Intermediate tier content which is in addition to foundation tier content is in underlined text. Higher tier content which is in addition to intermediate tier content is in bold text.