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
MathsUKwww.geekhero.co.uk
- 1.During one night on a mountain, four hikers each recorded the temperature they felt: Ben −6 °C, Priya −2 °C, Sam −9 °C, Alex 1 °C. Work out the difference between the coldest and the warmest of these four temperatures.
- 2.Write these numbers in order, starting with the smallest: −1.4, 5/4, −6/5, 1.3, 0
- 3.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.
- 4.Write down the symbol that makes this statement true: 3/10 ___ 0.3
- 5.Which of these numbers is the largest: −9, −7, −4, −2?
- 6.Write down the symbol that makes this statement true: 6/11 ___ 4/11
- 7.Which of these numbers lies between −8 and −5 on a number line?
- 8.Write these decimals in order, starting with the smallest: 0.6, 0.55, 0.601, 0.5, 0.56
- 9.Which one of these statements about negative numbers is true?
- 10.Write these numbers in order, starting with the largest: −0.6, 0.45, −0.15, 0.5, −0.09
- 11.A number line runs from 0 to 1 and is divided into 4 equal parts. Which mark on the line represents 1/4?
- 12.On a number line, point A is at −1. Point B is 4 units from point A. Write down the two possible positions of point B.
- 13.Amelia eats 5/8 of a bar of chocolate and Oliver eats 3/4 of an identical bar. Write both amounts as eighths and write down the greater of the two fractions.
- 14.n is a whole number and 2 ≤ n < 6. Write down all the possible values of n.
- 15.Write these numbers in order, starting with the smallest: 5, −8, 0, −3, 2
Answer key
- (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) −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) 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.
- (c) −2 — On a number line, negative numbers get smaller as their size (ignoring the sign) gets bigger, so −2 is closest to zero and is the largest of the four. A candidate who ignores the negative signs and orders the numbers as if they were positive, largest digit first, would pick −9 as the 'largest'. Continuing that same reversed ranking, the next number in digit order is −7. The number one step further along that same reversed ranking is −4, still short of the true largest value, −2.
- (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.
- (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 > −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.
- (c) 0.5, 0.45, −0.09, −0.15, −0.6 — Method: compare the decimals by their position on a number line, remembering that with negative decimals the one closer to zero is larger. Working: 0.5 and 0.45 are positive, so they come first, with 0.5 the larger of the two. Among the negatives, −0.09 is closest to zero, then −0.15, then −0.6 is furthest from zero and so the smallest. Answer: 0.5, 0.45, −0.09, −0.15, −0.6. 0.5, 0.45, −0.15, −0.09, −0.6 swaps −0.09 and −0.15, treating the negative decimal with more digits after the point as closer to zero. −0.6, −0.15, −0.09, 0.45, 0.5 lists the numbers from smallest to largest instead of largest to smallest. 0.5, 0.45, −0.6, −0.15, −0.09 orders the negative decimals by the size of the digit (0.6 > 0.15 > 0.09) as if they were positive, instead of recognising that a bigger negative decimal is further from zero and so smaller.
- (d) the first mark after 0 — Method: when a unit length is split into equal parts, each gap is one part of the whole, so 1/4 is one gap along from 0. Working: four equal parts means each gap measures 1/4, so the marks after 0 stand for 1/4, 2/4, 3/4 and 4/4. One gap along from 0 is therefore the mark for 1/4. Answer: the first mark after 0. The distractors: the second mark after 0 comes from counting 0 itself as the first mark; the third mark after 0 comes from counting back from the 1 end instead of forward from 0; the fourth mark after 0 comes from reading the 4 in the denominator as the number of the mark rather than the number of parts.
- (c) 3 or −5 — Method: a point a fixed distance from another can lie on either side of it, so move the given distance in each direction from the starting point. Working: moving 4 units to the right gives −1 + 4 = 3, and moving 4 units to the left gives −1 − 4 = −5. Answer: 3 or −5. The distractors: 5 or −3 comes from starting at 1 instead of −1, giving 1 + 4 and 1 − 4; 3 only comes from moving to the right and forgetting that the point could lie to the left as well; 4 or −4 comes from measuring the distance from zero instead of from point A, which just repeats the given distance.
- (c) 6/8 — Method: two fractions can only be compared directly when they share a denominator, so rewrite 3/4 in eighths and then compare the numerators. Working: 5/8 is already in eighths, and 3/4 = (3 × 2)/(4 × 2) = 6/8. Comparing the numerators, 6 > 5, so Oliver eats the larger share. Answer: 6/8. The distractors: 5/8 comes from skipping the conversion altogether and assuming that a bar cut into eighths must give the bigger share because it has more pieces; once both shares are written over the same denominator, 5 eighths is one eighth less than 6 eighths. 1/8 comes from working out how much more Oliver eats, 6/8 − 5/8, instead of writing down the greater of the two shares. 7/8 comes from adding 4 to the numerator and 4 to the denominator of 3/4 instead of multiplying both by 2.
- (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) −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.
Build your own mix at the worksheet builder.