Mathematics Year 7: Representing, modelling and connecting
Mathematics, Year 7 (Portugal): One relation told in words, table, graph and symbols; a model built from measurements; and a rectangle that connects geometry, numbers and algebra.
One relation, four ways of showing it
Suppose a sheet of paper weighs 5 g. In words: “the mass is 5 times the number of sheets”. In a table: 10 sheets → 50 g, 20 → 100 g, 30 → 150 g. In a graph: the points (10, 50), (20, 100) and (30, 150), on a straight line through the origin. In symbols: m = 5 × n, with n the number of sheets and m the mass in grams. Reading each one, and moving from one to another, is how you understand the relation. Say it in your own words too: “each sheet adds 5 g”.
Why symbols, and why keep the others
Each representation has its strength and its limit. The table is exact but lists only a few cases. The graph shows the whole relation at a glance, but exact values are hard to read. Words explain what the letters mean, but they are long. The symbols m = 5 × n are short, precise and predict any case: n = 1 200 gives m = 6 000 g. Without words you would not know what m and n stand for. That is why you keep switching, and why a spreadsheet or graphing app, which turns a formula into a table and a graph in a second, is so useful.
A case: a model made from measurements
Measurements: 10 sheets weigh 50 g, 40 sheets weigh 200 g and 100 sheets weigh 500 g. The mass per sheet is always 50 ÷ 10 = 200 ÷ 40 = 500 ÷ 100 = 5 g, so the model is m = 5 × n. Prediction: 300 sheets weigh 5 × 300 = 1 500 g, that is 1.5 kg. Use: a bag that carries at most 2.5 kg holds 2 500 ÷ 5 = 500 sheets. It is a model: real sheets differ a little, but it predicts well enough to plan, which is what a model is for.
+10% on one side and −10% on the other is not “no change”
A rectangle gets 10% longer and 10% narrower. Ana says the area stays the same, since “+10 and −10 cancel out”. Rui explains with a case: 20 cm by 10 cm has area 200 cm². Now 22 cm by 9 cm: 22 × 9 = 198 cm², a little smaller. With letters: 1.1 × l × 0.9 × w = 0.99 × l × w, so the area drops 1%. The 10% are of different quantities, so they do not cancel. Case, letters and geometry agree; listen to a counter-argument and test it with a number.
Where you see it: recipes, building, jobs
A recipe that uses 250 g of flour for 4 people uses 500 g for 8: mass and people are directly proportional, the same model as the sheets. Whoever scales a recipe, a dose or a plan uses it. Mathematics is also inside what people build: a tiled floor repeats a motif, and a carpenter works out lengths and angles before cutting a beam. Professionals use it to predict and to decide, and you can interview one to ask how.
Test what you learned
A table has n: 10, 20, 30 and m: 50, 100, 150. Which expression represents the same relation?
- a) m = n + 40
- b) m = 5 × n
- c) m = 50 × n
- d) n = 5 × m
Correct answer: b) m = 5 × n — Right: 5 × 10 = 50, 5 × 20 = 100 and 5 × 30 = 150, so all three pairs agree.
A sheet weighs 5 g, so m = 5 × n. A bag holds at most 4 kg. What is the largest number of sheets it can carry?
- a) 20
- b) 80
- c) 800
- d) 20 000
Correct answer: c) 800 — Right: 4 kg = 4 000 g, and 5 × n = 4 000 gives n = 4 000 ÷ 5 = 800.
A rectangle 20 cm by 10 cm becomes 10% longer and 10% narrower. What is its new area?
- a) 200 cm²
- b) 220 cm²
- c) 180 cm²
- d) 198 cm²
Correct answer: d) 198 cm² — Right: 22 cm × 9 cm = 198 cm², which is 99% of 200 cm². With letters, 1.1 × 0.9 = 0.99.
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