Communicate, represent and connect maths to the world — 11–13
Say the same idea in words, diagrams, numbers and symbols, explain your thinking, link topics together and use maths as a model of real situations.
One idea, many ways of saying it
The same idea can be shown in words, in a diagram, in a table or in symbols. "Three quarters of 12" is a sentence; it is also 12 squares in 4 groups of 3 with 3 groups shaded; it is 12 ÷ 4 × 3; it is 0.75 × 12. Reading these and converting from one to another is how you understand. And so is explaining your own way of thinking in your own words, listening to others and asking "why?" — that is how mistakes get found.
Why symbols: shorter and exact
"Twice n plus 3" can mean two things. For n = 5, either 2 × 5 + 3 = 13 or 2 × (5 + 3) = 16. In words the sentence is long and ambiguous; in symbols, 2 × n + 3 or 2 × (n + 3), nobody gets confused. Symbolic language is shorter, precise and the same for everyone. But symbols need words and diagrams to have meaning, and words need symbols to be exact: that is why you keep moving between them.
A case: 3/4 of 12 in five representations
How much is three quarters of 12? Words: split 12 into 4 equal parts and keep 3. Diagram: 12 squares in 4 groups of 3; shading 3 groups gives 9 squares. Symbols: 12 ÷ 4 = 3 and 3 × 3 = 9. Decimal: 3/4 = 3 ÷ 4 = 0.75 and 0.75 × 12 = 9. Percentage: 0.75 = 75% and 75% of 12 = 9. Five representations, the same 9, and they link topics: the fraction, the decimal and the percentage are the same number, and the diagram shows why.
A right answer can hide a wrong idea
Ana and Rui both say 3/4 of 12 is 9. Ana did 12 ÷ 4 × 3; Rui did 12 − 3. The answer alone hides the difference. Explaining shows it: for 3/4 of 20, Ana gets 20 ÷ 4 × 3 = 15 and Rui gets 20 − 3 = 17. Rui's method only agreed at 12 because 1/4 of 12 happens to be 3. Saying how you thought, listening to others and asking "and if the numbers change?" brings out mistakes that the answer hid.
Where you see it: a model of a train journey
A train runs at a constant 80 km/h. Table: 1 h gives 80 km, 2 h give 160, 3 h give 240, 4 h give 320. Model: distance = 80 × hours. Prediction: 5 h give 400 km. It is a model, since a real train stops and slows down, but it predicts and helps you plan. Maths is also in the patterns of a building's façade or a rose window, and in jobs like planning timetables or building: it helps to imagine and create what exists.
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