Flags Flags have been used for thousands of years. The first known use was by armies, as standards. Then they were used by sailors, at sea, and now they have many different purposes. Flags are usually designed so that they can be easily seen from a distance. They need to be recognised and understood even when they are being waved or moved, for example, in the wind. The designs of most flags include colour and many include symmetry. Task The table opposite shows some flags that are used as warnings on beaches in different European countries. For each of the flags in the table, find: a the number of lines of symmetry b the order of rotational symmetry. a Look at Flags 5, which are all shown the correct way up. Would you be able to tell if the lifeguard were to hang them upside down? Which ones would look different? b What do you notice about the order of rotational symmetry of those flags that would look the same if they were hung upside down? 3 This is a jellyfish warning flag, used on Spanish beaches. Write three sentences commenting on its symmetry and design. Explain whether you think its design is good or bad, giving reasons for your answer. 4 Learning objectives Representing Level : understand simple problems in familiar and unfamiliar situations Analysing Level : use mathematics to obtain answers to simple practical problems Interpreting Level : interpret and communicate solutions to practical problems, drawing simple conclusions and giving explanations links with Citizenship Art Science English
Flag Country and meaning Flag UK: Lifeguards patrolling Safe to swim Flag UK: Area used by watercraft (such as windsurfers and kayaks) Not safe to swim Flag 3 UK: Watercraft use prohibited Flag 4 Portugal: Beach temporarily without a lifeguard Flag 5 France: Windsurfing dangerous due to strong winds or sea conditions Flag 6 France: Swimming and use of floating devices hazardous because of land breeze Flag 7 France: Surfing area 43
Task Semaphore is a system of signalling, using the positions of flags. It is useful when electronic communications may be difficult, for example, in the mountains. Each letter of the alphabet is represented by two flags, one held in the signaller s right hand and one held in the left hand. For example, this is the signal for the letter P. Imagine that the signaller can only use the positions described. In each case, each letter is represented by two flags, one in the right hand and one in the left hand. Two right and two left a How many possible signals can the signaller make? b How many more letters than this are there in the alphabet? Three right and two left a How many possible signals can the signaller make now? b How many more letters than this are there in the alphabet? High 3 Four right and two left a How many possible signals can the signaller make now? b How many more letters than this are there in the alphabet? c What do you notice about the number of possible signals, compared to the numbers of right positions and left positions? These are all of the possible semaphore positions. Note: The down signal can only be made by one hand at a time (right or left). This is different from the up signal, which can be made by both hands (right and left). 4 How many different right positions, including down, are there? High Low High Low Down 5 How many different left positions, not including down, are there? 6 How many possible signals can the signaller make altogether? 7 What might you do to create the signals for the remaining letters in the alphabet? 8 Look at data sheet. What does the signaller do to create signals for the letters that have not been included in the signals you have seen? Hint: Look at letters H, I, O, W, X, Z. 9 Write two sentences to say whether you think the semaphore system is a good or a bad way of sending messages. Give reasons for your answer. 44
Task 3 (extension) Use semaphore to sketch a short message to a friend. Task 4 Look at data sheet, which gives the overall proportions of the national flags of the countries in the UK. You should see that the normal proportions for height to length are 3 : 5. Ella has a sheet of A4 paper. She wants to draw the flag for England to scale, but it won t fit. a She thinks if she divides all the real dimensions by the same number, she could make it fit, without making it too small. What number should she divide by? b Sketch the flag with the dimensions Ella will use on it. Lily has a sheet of A5 paper. She wants to draw the flag for Scotland to scale, but it won t fit. a What number could she divide the dimensions on the diagram by, to make it fit, without making it too small? b Sketch the flag with the dimensions Lily will use on it. 3 Mandy has a sheet of A3 paper. She wants to draw the Union flag with the dimensions exactly as on data sheet, in centimetres, but she needs some help. a She thinks it will be too difficult to draw an angle of 30.96. What angle do you think Mandy should draw to make it easier? b Mandy doesn t know how far along to draw the three stripes labelled, 6 and, below. They need to be in the middle of the flag. 50 6 Sketch this part of the flag with the dimensions on it to show Mandy how far along the stripes should be. c Mandy has the same trouble with these stripes labelled, 6 and, on the right. Again, she needs them to be in the middle of the flag. Sketch this part of the flag with the dimensions on it to show Mandy how far along the stripes should be. 30 6 45
4 Mark measures the picture of the Welsh flag on data sheet. He decides to enlarge it to A on a copy shop s photocopier. He wants to make it as large as possible, but doesn t want to lose any of the picture. Which option should Mark choose on the photocopier? 3 4 5 6 7 Explain your answer, giving the dimensions of Mark s flag. 5 Look very carefully at each flag. Write two sentences about each one, commenting on its lines of symmetry and order of rotational symmetry. Say whether you think it is a good or bad design for a flag, giving reasons for your answer. Task 5 Flagpoles are usually too high to measure simply and accurately, but this task shows you a way to estimate the height. You will need two metre rulers, a set square or a protractor and a sunny day! Find the shadow of the flagpole. Now ask a partner to hold a metre ruler so that it touches the ground, somewhere close to the flagpole. Use a set square or a protractor to make sure it is perpendicular to the ground. The shadow of the metre ruler and the shadow of the flagpole should be parallel. Use the other metre ruler to measure the lengths of the shadows of the ruler and the flagpole. The proportion of the heights to the shadows will be exactly the same: height of flagpole height of ruler = length of flagpole s shadow length of ruler s shadow Then you can work out the height of the flagpole: height of flagpole = length of flagpole s shadow Councillor Murray is doing a survey of the heights of flagpoles in his town, so he knows what lengths of rope will be required when the hoisting equipment needs replacing. He uses the method shown above and sketches his findings. Use the sketches to calculate the height of each flagpole. 46 height of ruler length of ruler s shadow
Flagpole Flagpole m.7 m Flagpole 3 m 0.6 m 0.9 m m. m m 0.5 m 5.4 m Flagpole 4 3.9 m 0.75 m Councillor Murray decides to write a report, including the heights flags should be raised on the flagpoles in different circumstances. For example, when a very important person dies, such as a member of the Royal Family, the flags should be lowered to half-mast. This means the top of the flag is two-thirds up the flagpole. How far up the flagpole is half-mast for each of the flags in question? Task 6 (extension) Ella, Lily and Mandy decide to make flagpoles out of bamboo canes and fly their flags (see Task 5) in the garden. They each have a choice of a 60 cm, 90 cm or 50 cm cane. Choose a cane for each girl s flag. Sketch each of the flags flying: at the top of its flagpole at half-mast. Include dimensions on your sketches. Task 7 A children s charity is running a competition for a well-designed flag to represent it. The flag will be flown on the rooftop of the charity s headquarters. It would also like the position of the flag on the flagpole to communicate a message. Design and draw a flag, including the dimensions. Then decide what message the flag could be used to send by its position on the flagpole. Write a paragraph to explain.? How did you find these tasks? What did you find easy or difficult about these tasks? What did you learn about how maths is used and applied in real-world situations? Did you work on your own, in pairs or in groups, and how did this help or hinder your approach and success with these tasks? 47