Contextualised task 39 Fun with Flags. Task 1: From fractions to ratios

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1 Contextualised task 39 Fun with Flags Teaching notes This task focuses on the mathematical specification of s. Students will first consider the of Wales, and then of the UK. They will see that there are very particular ways of defining a 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 s are constructed. They investigate the specifications for s 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

2 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 Slide 3 All s are constructed using precise mathematical information. A correct Welsh 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 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 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, the dragon is 8 units across. The box in which the dragon is drawn must be placed centrally on the.) Here is a diagram of the of England. Work with a partner to find 8 mathematical facts about this. You could give hints about ratio, area, enlargement For example: If the 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 is 1 metre wide, the area of white is = 4224 cm 2. The ratio of division along the top edge of the is 11:3:11. Ensure that examples such as those on the above list are discussed

3 Slide 4 The Union Flag (sometimes referred to as the Union Jack) is an attempt to combine the s of countries in the United Kingdom (The Union). Advance one click Here are the s of England, Scotland and Northern Ireland. At the time of the creation of the, (and the union of kingdoms) over 200 years ago, Wales was part of the Kingdom of England. Therefore, its own was not integrated. Slide 5 This is a complex 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 has an aspect ratio of 1:2. It is twice as long as it is wide. Most countries have a variant of their for particular purposes. The UK can also be constructed using an aspect ratio of 3:5. Advance one click This is the version of the union used by the army. This war is sometimes referred to as an instead of a.

4 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 is the ratio of width to length in its simplest form. Not all the s 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 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 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 s from each of Norway, Sweden, Finland, Denmark and Iceland. 1. Find each missing aspect ratio in the table. 2. The Danish and Icelandic s 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 could have a horizontal colour split defined: at the end of the diagonal cut (as with the Norwegian state 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

5 Flag Image Aspect ratio Sweden: state Notes 5:8 The colours are split 5:2:9 horizontally and 4:2:4 vertically Finland: civil Denmark: civil Norway: civil The colours are split 4:3:4 vertically and 5:3:10 horizontally 14:17 The white cross must be 1/7 of the '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 Koffardiet. Source: The colours are split 6:1:2:1:6 vertically and 6:1:2:1:12 horizontally Iceland: civil Sweden: naval Finland: naval Denmark: state Norway: state Iceland: state Task A: Mark scheme 18:25 The arms of the cross extend to the edge of the, and their combined width is 2/9, but the red cross 1/9 of the combined width of the. 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: 1:2 The colours are split 4:2:4 vertically and 5:2:5:8 The colours are split 4:3:4 vertically and 5:3:6:5 56:107 "The cross must be 1/7 of the 's height. The two first fields must be square in form with the height of 3/7 of the '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 Kongeet. Source: The colours are split 6:1:2:1:6 vertically and 6:1:2:1:6:11 9:16 The state 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:

6 The information below is intended as a guide only Full credit Finds the ratios as stated in the table below: Flag Image Aspect ratio Sweden: state Notes 5:8 The colours are split 5:2:9 horizontally and 4:2:4 vertically Finland: civil 11:18 The colours are split 4:3:4 vertically and 5:3:10 horizontally Denmark: civil 14:17 The colours are split 3:1:3 vertically and 6:2:9 horizontally Norway: civil 8:11 The colours are split 6:1:2:1:6 vertically and 6:1:2:1:12 horizontally Iceland: civil 18:25 The colours are split 7:1:2:1:7 vertically and 7:1:2:1:14 horizontally Sweden: naval Finland: naval 1:2 The colours are split 4:2:4 vertically and 5:2:5:8 11:19 The colours are split 4:3:4 vertically and 5:3:6:5 Denmark: state Norway: state 56:107 The colours are split 3:1:3 vertically and 24:8:75 16:27 The colours are split 6:1:2:1:6 vertically and 6:1:2:1:6:11 Iceland: state 9:16 The colours are split 6:1:2:1:6 vertically and 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 s on squared paper helpful, especially for the Icelandic s.

7 Note that in the case of the Danish state, 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 : 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.

8 Fun with Flags Task B: Question The information about the Danish and Icelandic s stated the official state guidelines and laws. Information about the s 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 ; The white cross must be 1/7 of the '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 s. The descriptions must use fractions. Your paragraphs are not allowed to contain ratios. Flag Image Aspect ratio Sweden: state Notes 5:8 The colours are split 5:2:9 horizontally and 4:2:4 vertically Finland: civil Norway: civil The colours are split 4:3:4 vertically and 5:3:10 horizontally The colours are split 6:1:2:1:6 vertically and 6:1:2:1:12 horizontally Sweden: naval Finland: naval Norway: state 1:2 The colours are split 4:2:4 vertically and 5:2:5:8 The colours are split 4:3:4 vertically and 5:3:6:5 The colours are split 6:1:2:1:6 vertically and 6:1:2:1:6:11

9 Task B: Mark scheme The information below is intended as a guide only Full credit Writes a paragraph to describe each. 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. Note also that some students may find using squared paper helpful. Swedish state An arm of the yellow cross has width 1/5 of the width of the. 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. Finnish civil An arm of the blue cross has width 3/11 of the width of the. 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 An arm of the blue-and-white cross has a width of 1/4 of the width of the, 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 An arm of the yellow cross has width 1/5 of the width of the. 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. Finnish naval An arm of the blue cross has width 3/11 of the width of the. 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. Norwegian civil An arm of the blue-and-white cross has a width of 1/4 of the width of the, 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.

10 Partial credit Completes a correct paragraph for four or five of the s Limited credit Completes a correct paragraph for two or three of the s No credit Any other response. Progression in reasoning Identify processes and connections transfer mathematical skills across the curriculum in a variety of contexts and everyday situations Apply skills within familiar contexts e.g. understands the meaning of an aspect ratio Identify, perhaps with some guidance, the skills needed within increasingly complex and unfamiliar contexts e.g. converts the ratios into fractions (task 2) with some guidance Identify independently the skills needed within increasingly complex and unfamiliar contexts e.g. converts the ratios into fractions (task 2) without guidance Represent and communicate explain results and procedures precisely using appropriate mathematical language Explanations are clear both orally and in writing, using some mathematical vocabulary. If written as instructions, they will lead to the intended correct result. e.g. writes a paragraph to describe construction of the Swedish state A wider range of appropriate mathematical vocabulary is used in explanations. Arguments are supported with evidence. e.g. writes a paragraph to describe the Swedish naval Orally and in writing: use mathematical vocabulary precisely e.g. uses fractions fluently to describe at least four of the s in task 2 Review select and apply appropriate checking strategies e.g. draws a rectangular on squared paper e.g. draws any on squared paper and labels distances to check against the ratios e.g. converts from fractions to ratios, and then checks by converting ratios back into fractions GCSE Content GCSE Mathematics Numeracy and GCSE Mathematics 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 fraction or percentage 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 scale drawings GCSE Mathematics only 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.