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.
Non-calculator
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 largest: 7/8, 0.8, 78%, 17/20
- 3.Which of these decimals lies between 1/4 and 2/5?
- 4.Write these numbers in order, starting with the smallest: 5, −8, 0, −3, 2
- 5.Write down the symbol that makes this statement true: −2/3 ___ −5/6
- 6.Write these three numbers in order, starting with the smallest: 3/5, 0.55, 58%
- 7.Which of these numbers is the largest: −9, −7, −4, −2?
- 8.A number line runs from 0 to 1 and is divided into 4 equal parts. Which mark on the line represents 1/4?
- 9.Write these numbers in order, starting with the largest: −0.6, 0.45, −0.15, 0.5, −0.09
- 10.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.
- 11.Four locations each record a temperature one winter morning. Which of these temperatures is the coldest?
- 12.Write these three numbers in order, starting with the smallest: 7/20, 0.3, 32%
- 13.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.
- 14.Write these numbers in order, starting with the smallest: −1.4, 5/4, −6/5, 1.3, 0
- 15.Which of these numbers lies between −4 and −1 on a number line?
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.
- (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) −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.
- (c) > — Method: write both fractions with a common denominator so they can be compared directly. Working: −2/3 = −4/6. Comparing −4/6 and −5/6, −4 is greater than −5, so −4/6 is greater than −5/6. Answer: −2/3 > −5/6, so the symbol is >. < comes from treating the fraction with the larger numerator and denominator, 5/6, as the greater number, ignoring that both fractions are negative. = comes from assuming the fractions must be equal because they look similar in size. ≤ combines < (false) with = (false).
- (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) −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.
- (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) 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.
- (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) −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.
- (b) 0.3, 32%, 7/20 — Method: convert every number to a decimal so they can be compared on the same scale. Working: 7/20 = 0.35, 0.3 stays as 0.3, and 32% = 0.32. Comparing 0.3, 0.32 and 0.35 in size gives the order 0.3, then 0.32, then 0.35. Answer: 0.3, 32%, 7/20. 7/20, 32%, 0.3 lists the numbers from largest to smallest instead of smallest to largest. 0.3, 7/20, 32% swaps 32% and 7/20, treating the fraction 7/20 as smaller even though 7/20 = 0.35 is bigger than 32% = 0.32. 32%, 0.3, 7/20 comes from moving the digits one place too far when converting the percentage, giving 0.032 instead of 0.32, which makes 32% look far smaller than it really is.
- (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.
- (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.
- (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.
Build your own mix at the worksheet builder.