Printable · GCSE Foundation · ages 14-16
Ordering numbers and inequality symbols worksheet — GCSE Foundation
Fifteen questions on "ordering numbers and inequality symbols" — DfE statement N1. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Answer key: Ordering numbers and inequality symbols worksheet — GCSE Foundation
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- (d) −2 — Method: place the two end values on a number line and list the integers that sit strictly between them. Working: reading from left to right the integers run −4, −3, −2, −1, so the values strictly between the ends are −3 and −2. Only one of those is offered. Answer: −2. The distractors: −5 comes from ordering negatives by the size of their digits, which wrongly places −5 to the right of −4; 0 comes from carrying on past −1 instead of stopping at it; 2 comes from ignoring the minus signs and choosing a number between 1 and 4.
- (b) −0.7 < −0.25 — Method: compare the two negative decimals by their distance from zero on a number line. Working: −0.7 is 0.7 away from zero and −0.25 is 0.25 away from zero, so −0.7 is further from zero in the negative direction, making it the smaller number. Answer: −0.7 < −0.25 is true. "−0.7 > −0.25" comes from comparing 0.7 and 0.25 as if both numbers were positive, ignoring the negative signs. "−0.7 = −0.25" comes from assuming the two numbers are equal because they are both negative decimals. "−0.7 ≥ −0.25" combines the false statement "−0.7 > −0.25" with the false statement "−0.7 = −0.25".
- (b) 0 > −1 — Method: test each statement by placing both of its numbers on a number line; the greater number is the one further to the right. Working: every negative number lies to the left of zero, so zero is greater than −1. Among the negatives, −3 lies to the right of −5, so −5 is not greater than −3. Answer: 0 > −1. The distractors: −5 > −3 comes from ordering the negatives by the size of their digits, so that 5 makes −5 look the larger; −3 > 0 comes from ignoring the minus sign and comparing 3 with 0; 0 < −1 comes from the belief that zero is the smallest number there is, so that even a negative number is above it.
- (a) −4.5 °C — Order the temperatures by their actual value on a number line, remembering that a more negative number is further below zero and therefore colder: −4.5 °C is the coldest, since it is further below zero than −4.05 °C, −3.8 °C or 2 °C. Comparing the digits 405 and 45 as though the decimal points lined up, without padding −4.5 to match the number of decimal places in −4.05 first, makes −4.05 °C look like it has the bigger size, so it gets picked as the coldest by mistake — in fact −4.05 °C is closer to zero than −4.5 °C, not further from it. Picking −3.8 °C comes from choosing the negative reading with the smallest absolute value, forgetting that for negative numbers, a smaller absolute value means a warmer, less negative temperature, not a colder one. Picking 2 °C comes from ignoring the negative signs on the other three readings altogether and comparing raw digit sizes, when in fact any negative temperature is colder than any positive temperature. So the coldest temperature is −4.5 °C.
- (a) 2, 3, 4, 5 — Method: work out which whole numbers satisfy both parts of the inequality. Working: n ≥ 2 means n can be 2 or more; n < 6 means n must be less than 6, so 6 itself is not included. The whole numbers that fit both conditions are 2, 3, 4 and 5. Answer: 2, 3, 4, 5. 2, 3, 4, 5, 6 treats < 6 as ≤ 6 and wrongly includes 6. 3, 4, 5 treats ≥ 2 as > 2 and wrongly leaves out 2. 1, 2, 3, 4, 5 wrongly includes 1, which does not satisfy n ≥ 2.
- (c) < — Method: compare the two negative numbers by their position on a number line, not by the size of the digit. Working: −9 is nine places to the left of zero and −4 is four places to the left of zero, so −9 is further from zero in the negative direction, which makes it the smaller number. Answer: −9 < −4, so the symbol is <. > comes from comparing 9 and 4 as if both numbers were positive, ignoring the negative signs. = comes from treating both numbers as equal simply because they are both negative. ≥ combines > (false) with = (also false).
- (a) −15 °C — Method: the lowest temperature is the reading furthest to the left on a number line, and among negative readings that is the one furthest below zero. Working: the three negative readings are 1 degree, 7 degrees and 15 degrees below zero, and 0 °C is not below zero at all. Fifteen degrees below zero is the furthest below. Answer: −15 °C. The distractors: −1 °C comes from ordering the negatives by the size of their digits, so that the smallest digit is taken as the lowest temperature; −7 °C comes from comparing 15 and 7 by their leading digits, deciding that 1 is less than 7 and so that −7 lies below −15; 0 °C comes from treating zero as the smallest number there is.
- (d) −8, −3, 0, 2, 5 — Method: place the numbers on a number line and read them from left to right. Working: the negative numbers −8 and −3 come before 0, with −8 further left than −3 because it is further from zero in the negative direction; then 0, then the positive numbers 2 and 5 in increasing size. Answer: −8, −3, 0, 2, 5. −3, −8, 0, 2, 5 swaps −3 and −8, putting the negative number closer to zero first. 5, 2, 0, −3, −8 lists the numbers from largest to smallest instead of smallest to largest. −8, −3, 2, 0, 5 puts 2 before 0, treating positive numbers as if they always come before zero.
- (a) 7/8, 17/20, 0.8, 78% — Method: convert every value to a decimal, then order the decimals from largest to smallest. Working: 7/8 = 0.875, 17/20 = 0.85, 0.8 = 0.8, 78% = 0.78. Ordering from largest to smallest gives 7/8, 17/20, 0.8, 78%. Answer: 7/8, 17/20, 0.8, 78%. '78%, 0.8, 17/20, 7/8' comes from ordering the converted decimals from smallest to largest instead of largest to smallest. '17/20, 7/8, 0.8, 78%' comes from converting 17/20 incorrectly as larger than 7/8, for example treating 17/20 as 0.87 instead of 0.85, swapping the top two. '7/8, 0.8, 17/20, 78%' comes from converting 17/20 incorrectly as equal to 0.8, swapping the middle two.
- (d) −5, −1, 0, 3 — Method: order the numbers by their position on a number line, smallest (furthest left) first. Working: both −5 and −1 lie to the left of 0, and 3 lies to the right of 0. Of the two negatives, −5 is 5 units from zero and −1 is 1 unit from zero, so −5 is further left. Answer: −5, −1, 0, 3. The distractors: 3, 0, −1, −5 is the correct order written the wrong way round, starting with the largest; −1, −5, 0, 3 comes from ordering the two negatives by the size of their digits, so that −1 is treated as the smaller; 0, −1, −5, 3 comes from believing that zero is the smallest number there is and then listing the negatives by their digits.
- (c) < — Compare the two decimals by their value, not by how many digits they have: 0.45 is worth less than half, while 0.5 is exactly half, so 0.45 is smaller. The correct symbol is <, since 0.45 is less than 0.5. Choosing > treats 0.45 as bigger because it has more digits after the decimal point than 0.5 — extra decimal digits do not make a number bigger. Choosing = comes from rounding 0.45 to 1 decimal place, 0.5, and then treating the rounded value as if it were the original number. Choosing ≥ would mean 0.45 is greater than or equal to 0.5, which is false in both parts, since 0.45 is neither equal to nor bigger than 0.5. So 0.45 < 0.5.
- (a) 10 °C — Method: work out the coldest and warmest of the four temperatures, then subtract to find the difference. Working: the coldest temperature is Sam's, −9 °C, and the warmest is Alex's, 1 °C. The difference is 1 − (−9) = 1 + 9 = 10. Answer: 10 °C. 8 °C comes from working out 1 − 9 = −8 and reporting 8, dropping the negative sign on −9 instead of turning the subtraction into an addition. 3 °C comes from comparing the wrong pair, Sam's −9 °C and Ben's −6 °C, instead of the coldest and the warmest: −6 − (−9) = 3. 7 °C comes from comparing Ben's −6 °C with Alex's 1 °C, mistakenly treating Ben's reading as the coldest instead of Sam's.
- (d) 0.3 — Converting the fractions to decimals, 1/4 = 0.25 and 2/5 = 0.4, so any decimal between 0.25 and 0.4 is a valid answer, and 0.3 fits. Confusing 1/4 with 1/5 and converting it as 0.2 instead of 0.25 gives a value below the true lower bound. Confusing 2/5 with 1/2 and converting it as 0.5 instead of 0.4 gives a value above the true upper bound. Converting the fractions correctly but choosing a decimal above the true upper bound of 0.4 instead of between the two values gives 0.45.
- (d) −1.4, −6/5, 0, 5/4, 1.3 — Method: convert the fractions 5/4 and −6/5 to decimals so every number is written the same way, then compare all five decimals. Working: 5/4 = 1.25 and −6/5 = −1.2. Comparing −1.4, −1.2, 0, 1.25 and 1.3 in size gives the order −1.4, −1.2, 0, 1.25, 1.3. Answer: −1.4, −6/5, 0, 5/4, 1.3. −6/5, −1.4, 0, 5/4, 1.3 swaps the two negative numbers, treating −6/5 as more negative than −1.4 even though −1.2 is closer to zero than −1.4. 1.3, 5/4, 0, −6/5, −1.4 lists the numbers from largest to smallest instead of smallest to largest. −1.4, −6/5, 0, 1.3, 5/4 swaps 5/4 and 1.3, comparing the numerator 5 directly with 1.3 instead of converting 5/4 to the decimal 1.25 first.
- (a) 0.5, 0.55, 0.56, 0.6, 0.601 — Compare the decimals by giving them all the same number of decimal places first: 0.600, 0.550, 0.601, 0.500, 0.560. Ordering these from smallest to largest gives 0.500, 0.550, 0.560, 0.600, 0.601, which is 0.5, 0.55, 0.56, 0.6, 0.601. Comparing the digits as though they were whole numbers, reading 0.601 as "601" and 0.5 as "5", without padding to the same number of decimal places, gives the wrong order 0.5, 0.6, 0.55, 0.56, 0.601, because it ignores the place value of each digit. Ordering largest to smallest instead of smallest to largest, as the question asks, gives 0.601, 0.6, 0.56, 0.55, 0.5. Misreading the close values 0.55 and 0.56 and swapping them gives 0.5, 0.56, 0.55, 0.6, 0.601. So the correct order, smallest to largest, is 0.5, 0.55, 0.56, 0.6, 0.601.
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