Foundation November 2019 Paper 2 Q20
20 A map of an island is shown on a centimetre grid.
\(A\), \(B\) and \(C\) are houses.
(a) The actual distance between \(A\) and \(B\) is 1500 metres.
Show that the scale on the map is 1 : 30 000 [2 marks]
(b) Work out the actual distance between \(A\) and \(C\).
Give your answer in kilometres. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 shown by valid calculation | ||
| \(1500 \times 100\) or \(30\,000 \times 5\) or \(1500 \div 5\) or \(30\,000 \div 100\) or \(5 \div 100\) or \(1500 \times 100 \div 5\) or \(30\,000 \times 5 \div 100\) or \(1500 \times 100 \div 30\,000\) | M1 | must see one of these calculations but may evaluate incorrectly for M1 do not allow embedded in an invalid calculation eg \(30\,000 \times 5 \div 1000\) is M0 |
| \(\dfrac{1500 \times 100}{5} = 30\,000\) or \(\dfrac{30\,000 \times 5}{100} = 1500\) or \(\dfrac{1500 \times 100}{30\,000} = 5\) and \(AB = 5\) or \(1500 \times 100 = 30\,000 \times 5\) or \(1500 \div 5 = 30\,000 \div 100\) | A1 | must show correct use of all four of 1500, 100, 5 and 30 000 may be in two stages eg \(1500 \times 100 = 150\,000\) and \(150\,000 \div 5 = 30\,000\) or \(1500 \div 5 = 300\) and \(30\,000 \div 100 = 300\) if units shown must be correct for A1 |
| Alternative method 2 shown by unit conversion and valid calculation | ||
| 150 000 cm or 300 m or 0.05 m | M1 | correct units must be shown to imply use of 100 |
| 150 000 cm and \(30\,000 \times 5 = 150\,000\) or 150 000 cm and \(150\,000 \div 5 = 30\,000\) or 150 000 cm and \(150\,000 \div 30\,000 = 5\) and \(AB = 5\) or 30 000 cm and 300 m and \(1500 \div 5 = 300\) or 30 000 cm and 300 m and \(300 \times 5 = 1500\) or 30 000 cm and 300 m and \(1500 \div 300 = 5\) and \(AB = 5\) or 0.05 m and \(1500 \div 0.05 = 30\,000\) or 0.05 m and \(30\,000 \times 0.05 = 1500\) | A1 | correct units must be shown |
Additional guidance
\(30\,000 \times 5\) may be seen as a correct build-up ie 30 000, 60 000, 90 000, 120 000, 150 000
Measuring \(AB\) as a value other than 5 will score M1 max
Using \(AC\) or \(BC\) can only score a max of M1 for one of the calculations or conversions that does not use \(AB\)
Allow M1 even if seen among other incorrect work but for A1 their method must be all correct and unambiguous
Must show a calculation from Alt 1 or a value with units from Alt 2 for the M1 ie 150 000 only or 300 only or 0.05 only is M0
Ignore any additional reference to the grid having 100 squares
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 working in cm | ||
| [4.4, 4.6] | B1 | may be on diagram |
| their \([4.4, 4.6] \times 30\,000\) or [132 000, 138 000] | M1 | their \(AC\) must be in the range [4, 7] and must not be 5 [132 000, 138 000] implies B1M1 if no measurement for \(AC\) given |
| their [132 000, 138 000] \(\div 100 \div 1000\) | M1dep | oe must be converting into km |
| [1.32, 1.38] | A1ft | ft B0M2 |
| Alternative method 2 working in cm | ||
| [4.4, 4.6] | B1 | may be on diagram |
| \(\dfrac{\text{their } [4.4, 4.6]}{5} \times 1500\) or their \([4.4, 4.6] \times 300\) or [1320, 1380] | M1 | their \(AC\) must be in the range [4, 7] and must not be 5 [1320, 1380] implies B1M1 if no measurement for \(AC\) given |
| their [1320, 1380] \(\div 1000\) | M1dep | oe must be converting into km |
| [1.32, 1.38] | A1ft | ft B0M2 |
| Alternative method 3 working in mm | ||
| [44, 46] | B1 | may be on diagram |
| their \([44, 46] \times 30\,000\) or [1 320 000, 1 380 000] or \(\dfrac{\text{their } [44, 46]}{50} \times 1500\) or their \([44, 46] \times 30\) or [1320, 1380] | M1 | their \(AC\) must be in the range [40, 70] and must not be 50 [1 320 000, 1 380 000] implies B1M1 if no measurement for \(AC\) given [1320, 1380] implies B1M1 if no measurement for \(AC\) given |
| their [1 320 000, 1 380 000] \(\div 10 \div 100 \div 1000\) or their [1320, 1380] \(\div 1000\) | M1dep | oe must be converting into km |
| [1.32, 1.38] | A1ft | ft B0M2 |
Additional guidance
| Answer only in range [1.32, 1.38] | B1M1M1A1 |
| Answer must match their \(AC\) if seen | |
| Must be using the scale 1 : 30 000 or 5 : 1500 | |
| Their [4.4, 4.6] is often 4 (perhaps counting squares crossed diagonally) or 6 (perhaps 2 down and 4 across) | |
| 4 seen and answer 1.2 | B0M1M1A1ft |
| 4 seen and 120 000 (by Alt 1) or 4 seen and 1200 (by Alt 2) | B0M1M0A0 |
| Answer 1.2 (without 4 seen) | Zero |
| 6 seen and answer 1.8 | B0M1M1A1ft |
| 6 seen and 180 000 (by Alt 1) or 6 seen and 1800 (by Alt 2) | B0M1M0A0 |
| Answer 1.8 (without 6 seen) | Zero |
| 4.7 seen and answer 1.41 | B0M1M1A1ft |
| 4.7 seen and 141 000 (by Alt 1) or 4.7 seen and 1410 (by Alt 2) | B0M1M0A0 |
| Answer 1.41 (without 4.7 seen) | Zero |
| Using Pythagoras gives \(AC = \sqrt{20}\) or \(2\sqrt{5}\) or 4.4(72…) or 4.5 | B1 |