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.
Ordering numbers and inequality symbols worksheet — GCSE Foundation
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- 1.A weather app records the temperature at three points in one day: 6 °C at noon, −2 °C at midnight, and −7 °C just before dawn. Work out the difference between the highest and lowest of these three temperatures.
- 2.Write down the symbol that makes this statement true: 3/10 ___ 0.3
- 3.n is a whole number and 2 ≤ n < 6. Write down all the possible values of n.
- 4.Which of these numbers lies between −8 and −5 on a number line?
- 5.Write these decimals in order, starting with the smallest: 0.6, 0.55, 0.601, 0.5, 0.56
- 6.Which of these statements about −0.7 and −0.25 is true?
- 7.Write these three numbers in order, starting with the smallest: 3/5, 0.55, 58%
- 8.Write down the symbol that makes this statement true: −9 ___ −4
- 9.On one night in Manchester the temperature at 3 am was 8 °C below zero and at 9 am it was 3 °C below zero. Write down the warmer of the two readings.
- 10.Write down the symbol that makes this statement true: 6/11 ___ 4/11
- 11.Four locations each record a temperature one winter morning. Which of these temperatures is the coldest?
- 12.Write these numbers in order, starting with the largest: 7/8, 0.8, 78%, 17/20
- 13.Which of these decimals lies between 1/4 and 2/5?
- 14.Write these numbers in order, starting with the smallest: 3, −1, 0, −5
- 15.Write down the symbol that makes this statement true: 0.45 ___ 0.5
Answer key
- (a) 13 °C — Method: subtract the lowest temperature from the highest temperature to find the difference. Working: the highest temperature is 6 °C and the lowest is −7 °C. Difference = 6 − (−7) = 6 + 7 = 13. Answer: 13 °C. 8 °C comes from using −2 °C as the lowest temperature instead of −7 °C: 6 − (−2) = 8. 5 °C comes from finding the difference between the two negative temperatures instead of the highest and lowest: −2 − (−7) = 5. −1 °C comes from adding the highest and lowest temperatures instead of subtracting: 6 + (−7) = −1.
- (a) = — Method: convert the fraction to a decimal so both numbers are in the same form. Working: 3/10 means 3 divided by 10, which is 0.3. Answer: 3/10 = 0.3, so the symbol is =. < comes from treating the fraction as smaller than the decimal without converting it first. > comes from the same mistake the other way round. ≠ comes from assuming a fraction and a decimal can never be equal.
- (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.
- (d) −6 — Method: check each option to see if it falls strictly between −8 and −5, remembering that numbers become smaller as they get more negative. Working: −8 < −6 < −5, so −6 lies between them. Answer: −6. −9 comes from a value that is more negative than −8, so it lies outside the range, below −8. −4 comes from a value that is less negative than −5, so it lies outside the range, above −5. 6 comes from ignoring the negative signs and picking a positive number between 5 and 8.
- (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.
- (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".
- (a) 0.55, 58%, 3/5 — Converting all three to decimals: 3/5 = 0.6, 0.55 stays as 0.55, and 58% = 0.58. In order from smallest to largest, this is 0.55, then 58%, then 3/5. Writing the numbers in the reverse order, largest to smallest, gives 3/5, 58%, 0.55. Misconverting 3/5 as 0.5 instead of 0.6 makes it appear smaller than both other values, giving the order 3/5, 0.55, 58%. Misconverting 58% as 0.058 instead of 0.58, by moving the decimal point two extra places, makes it appear smallest of the three, giving the order 58%, 0.55, 3/5.
- (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) −3 °C — Method: write each reading as a signed temperature, then choose the one further to the right on a number line. Working: 8 °C below zero is −8 °C and 3 °C below zero is −3 °C. On a number line −3 lies to the right of −8, so it is the warmer reading. Answer: −3 °C. The distractors: −8 °C comes from ordering negatives by the size of their digits, treating −8 as the larger number; 3 °C has the right size but the sign dropped, and a reading of 3 °C is above zero rather than below it; 5 °C comes from working out the difference between the two readings instead of choosing one of them.
- (c) > — Method: fractions with the same denominator are made of parts of the same size, so compare how many of those parts each fraction has. Working: both fractions are elevenths, and 6 elevenths is 2 more elevenths than 4 elevenths, so the fraction on the left is the larger one. The symbol must have its point facing the smaller side. Answer: >. The distractors: < comes from comparing the numerators the wrong way round, as though a larger numerator gave a smaller fraction; = comes from seeing the same denominator in both fractions and concluding that the fractions themselves are the same size; ≤ comes from working out the direction correctly but then picking the wrong symbol for it, reading it as though it meant 'is greater than or equal to'; ≤ means 'is less than or equal to', and 6 elevenths is neither less than nor equal to 4 elevenths.
- (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) 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) 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) −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.
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