Mathematics Standard • Year 12 • Ratios and rates • Lesson 4
Estimating and Reversing, Mastery Challenge
Run the rule backwards to find a missing offset, compare one application against two on the same block, reverse a rainfall to find a roof area, and catch the unit error that costs the most marks in this topic.
1. Find the missing offset
Substitute what you know and solve for what you do not. 3 marks
Q1.1 One application of the trapezoidal rule gives an area of 240 square metres. The strip width is 8 m and the first offset is 22 m. Find the last offset.
2. One application against two
Same block, two different estimates. Part (c) is where the marks separate. 4 marks
Q2.1 A block is 20 m long. Offsets are measured at 10 m, 18 m and 12 m.
(a) Estimate the area using ONE application, using only the two end offsets. (b) Estimate it again using TWO applications. (c) State the difference and explain which estimate you would trust, and why.
3. Reverse a rainfall
You are given the water and asked for the roof. 4 marks
Q3.1 A tank collected 8400 L of water from a flat roof after 28 mm of rain. Find the area of the roof, and state any assumption you have made.
4. Will the tank hold it?
More than one quantity to keep track of. 5 marks
Q4.1 A shed roof measures 12 m by 6.5 m and drains into a 2000 L tank that is already 40 percent full. A storm drops 45 mm of rain. Does the tank overflow, and if so by how much?
5. Find the error
Identify the mistake, name it, and give the correct answer. 3 marks
Q5.1 A student writes: "The area is 400 square metres and 30 mm of rain fell, so the volume is 400 × 30 = 12 000 cubic metres." Explain what has gone wrong and give the correct answer.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, The missing offset
Substitute into A = (h/2)(df + dl): 240 = (8/2)(22 + dl), so 240 = 4(22 + dl) [1]. Divide both sides by 4: 60 = 22 + dl [1]. So dl = 38 m [1]. Check: (8/2)(22 + 38) = 4 × 60 = 240 square metres.
Q2.1, One application against two
(a) Using only the ends, the strip is the whole 20 m: A = (20/2)(10 + 12) = 10 × 22 = 220 square metres [1].
(b) With three offsets there are two strips, so h = 10 m: A = (10/2)(10 + 2 × 18 + 12) = 5 × 58 = 290 square metres [1].
(c) The difference is 290 − 220 = 70 square metres [1]. The two-application estimate is the one to trust. The middle offset of 18 m is much larger than either end, which means the boundary bulges outwards in the middle. One application draws a single straight line from end to end and misses that bulge entirely, so it under-counts. Narrower strips follow the real boundary more closely [1].
Q3.1, Reverse a rainfall
Convert the water to cubic metres: 8400 L = 8.4 cubic metres [1]. Convert the depth: 28 mm = 0.028 m [1]. Rearrange V = A × d to get A = V ÷ d = 8.4 ÷ 0.028 = 300 square metres [1]. Check: 300 × 0.028 = 8.4 cubic metres = 8400 L. Assumption: every drop that landed on the roof reached the tank, with nothing lost to splashing, leaks, overflow or evaporation [1].
Q4.1, Will the tank hold it?
Roof area = 12 × 6.5 = 78 square metres [1]. Depth = 45 mm = 0.045 m [1]. Volume of rain = 78 × 0.045 = 3.51 cubic metres = 3510 L [1]. The tank already holds 40 percent of 2000 L, which is 800 L [1]. Total water = 800 + 3510 = 4310 L, and the tank only holds 2000 L, so yes it overflows, by 4310 − 2000 = 2310 L [1].
Q5.1, Find the error
The depth was never converted from millimetres to metres [1]. Multiplying square metres by millimetres is not a valid calculation, and it inflates the answer by a factor of 1000. Correctly, 30 mm = 0.03 m, so V = 400 × 0.03 = 12 cubic metres [1], which is 12 000 litres [1].
Note the trap. The student's numeral, 12 000, is the right number attached to the wrong unit. An answer that looks familiar is not the same as an answer that is right, so always carry the unit through every line.