Math Detours · Grade 5 · Connections · 8

Fractions and symmetry in flags

Halves, thirds and quarters, mirror lines and half turns. The world's flags are a geometry lesson in colour.

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Tonight's idea

Many flags are built with two simple tools: equal parts and symmetry.

Equal parts. France's flag is three equal bands. Each colour covers \(\tfrac13\) of the flag.

Symmetry. A mirror line is a fold line. Fold along it and the two halves match exactly. A flag has half-turn symmetry if it looks the same after you turn it upside down.

Dashed lines are mirror lines. Japan is white with a red disc. Denmark is red with a white cross. Trinidad and Tobago is red with a black band edged in white. These are sketches, not exact.

France has one mirror line. Folding left over right fails, because blue lands on red. Japan has two mirror lines and half-turn symmetry. Denmark's cross sits to the left, so it has one mirror line. Trinidad and Tobago has no mirror line at all. Yet a half turn leaves it unchanged.

Watch one: Colombia

Colombia's flag has three stripes across. Yellow fills the top half. Blue and red share the bottom half equally.

So yellow is \(\tfrac12\), blue is \(\tfrac14\) and red is \(\tfrac14\). Check: \(\tfrac12 + \tfrac14 + \tfrac14 = 1\), the whole flag.

On a flag 60 cm high, yellow is 30 cm, blue 15 and red 15. There is one mirror line, from top to bottom. A half turn fails: yellow ends up at the bottom.

Your turn

Germany's flag is three equal stripes across: black, red and gold. One flag is 90 cm high and 150 cm wide. (a) What fraction of it is red? (b) How high is each stripe? (c) What is the area of the red stripe?
Mauritius has four equal stripes across: red, blue, yellow and green. Nigeria has three equal upright bands: green, white, green. (a) What fraction of the Mauritius flag is not red? (b) What fraction of the Nigeria flag is green? (c) How many mirror lines does each flag have?
Thailand has five stripes across: red, white, blue, white, red. The blue stripe is twice as high as each of the others. (a) Sketch it. Cut the flag into 6 equal strips to find the fraction for each colour. (b) How many mirror lines does it have? Does it have half-turn symmetry?
Switzerland's flag is a square. It is red with a white plus sign in the centre. (a) How many mirror lines does it have? (b) Japan's red disc alone has more mirror lines than you could count. Why does Japan's flag have only two?

Stretch: two folds make a turn

Japan, Nigeria and Thailand have two mirror lines each. All three have half-turn symmetry. Is that luck? Flip a flag left to right, then top to bottom. Where does the top-left corner go? And does half-turn symmetry always bring a mirror line?

Answers

(a) \(\tfrac13\). (b) \(90 \div 3 = 30\) cm. (c) \(30 \times 150 = 4{,}500\) cm².
(a) \(\tfrac34\). (b) \(\tfrac23\). (c) Mauritius has one, running from top to bottom. A fold the other way puts red on green. Nigeria has two: one from top to bottom through the white band, and one across the middle.
(a) The strips are red, white, blue, blue, white, red. Red is \(\tfrac26 = \tfrac13\), white is \(\tfrac26 = \tfrac13\) and blue is \(\tfrac26 = \tfrac13\). Each colour covers the same amount. (b) Two mirror lines: one from top to bottom, and one across through the middle of the blue. Yes, it has half-turn symmetry.
(a) Four: one across, one from top to bottom, and the two diagonals. (b) A fold must match the whole flag, edges and all. Japan's flag is a rectangle that is longer than it is high. A diagonal fold does not bring its edges together. Only the fold across and the fold from top to bottom do.

Stretch. It is not luck. The first flip sends the top-left corner to the top right. The second sends it to the bottom right. A half turn does exactly that. So the two flips together make a half turn. A flag with both mirror lines must have half-turn symmetry. The other way round fails: Trinidad and Tobago has half-turn symmetry and no mirror line.

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