Communicate, represent and connect — Mathematics, 11–13
One idea in words, diagram, table and symbols; explain your thinking and answer an argument — and see the same maths in juice, rose windows and surveys.
Words, diagram, table and symbols
A juice is made with 1 cup of concentrate for 4 cups of water. In words: “for each cup of concentrate, four of water”. In a diagram: a row of 1 square and 4 circles, repeated. In a table: concentrate 1, 2, 3 and water 4, 8, 12. In symbols: a = 4 × c, with c the cups of concentrate and a the cups of water. Reading each one and moving from one to another is how you understand the idea; and explaining your own way of thinking in your own words is the first step to discussing it.
Why symbols save words
Say “the number of cups of water is always four times the number of cups of concentrate”, then write a = 4 × c. Symbols are shorter and the same for everyone. They also let you predict: with c = 7, a = 4 × 7 = 28. But without words you do not know what c and a mean, and without symbols the words leave doubts. This juice also links topics: the ratio of 1 to 4, the table, a fraction of the total and a percentage are the same idea seen from different sides.
A case: 7 cups of concentrate
Rule: 1 cup of concentrate for 4 of water. Table: c = 1, 2, 3, …, 7 and a = 4, 8, 12, …, 28. Symbols: a = 4 × c, so 4 × 7 = 28 cups of water. Total: 7 + 28 = 35 cups of juice. What part of the juice is concentrate? 7 out of 35, that is 7/35 = 1/5 = 0.2 = 20%. Check with another case: c = 3 gives 12 of water, a total of 15, and 3/15 = 1/5 too. Two representations agree, so you can trust the result, and the diagram of one “square + 4 circles” block shows why every block has 1 part in 5.
“1 to 4” is not “a quarter”
Ana says: “1 of concentrate for 4 of water, so the concentrate is 1/4 of the juice.” Rui disagrees and explains with a case: “make 1 cup of concentrate and 4 of water; that is 5 cups, and the concentrate is 1 in 5.” Ana listened, asked “and with 2 cups?” and saw: 2 and 8 make 10 cups, and 2/10 = 1/5. The mistake is common: 4 is the water, not the whole. A badly worded sentence misleads, so describe your reasoning with numbers and listen to other people's arguments.
Where you see it: rose windows and surveys
In a rose window with 6 identical petals, each petal lands on the next when you turn it by 60°, because 360° ÷ 6 = 60°. With 8 petals, 45°. Architects and craftspeople use this to draw rose windows and patterns on façades. And if “3 in every 10” pupils of a school of 500 walk, the model gives 500 ÷ 10 × 3 = 150 pupils: a useful prediction for planning, even knowing it is an approximation. Maths helps you interpret the world and helps build it.
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