A number pattern can look simple until the rule changes just enough to trap an automatic answer. Suppose the terms are 2, 5, 9, 14. A pupil who notices that the numbers increase may add five again and write 19. But the gaps are 3, 4 and 5, so the next gap may be 6, giving 20. The important work is not the final addition. It is identifying exactly what is changing, and checking that the chosen relationship is supported by the terms given.
This Primary 6 guide is about one examination-performance job: when a sequence appears in a word problem, table, diagram or number row, read the terms by position, expose what happens between them, select a defensible rule, then check the requested term or missing value. You will also learn why a difference table is a tool rather than magic. A short list of numbers does not prove that only one mathematical rule is possible. In assessment questions, wording, a visible growing shape or another condition normally helps select the intended rule.
The core move: put the gaps in their own row
Write the terms in order, with equal spaces. Beneath the first and second terms, record the second minus the first. Continue between each pair. This new row is called the first differences. For 2, 5, 9, 14, write 3, 4, 5 below the gaps. The consecutive differences rise by one, suggesting that the next gap is 6. Therefore 14 + 6 = 20. Verify backwards: 20 – 14 = 6, and the gaps 3, 4, 5, 6 follow the stated increasing pattern.
For 6, 11, 16, 21, the gaps are all 5. That means each next term adds the same amount. If the question asks for the tenth term, there are nine jumps from term 1 to term 10. The tenth term is 6 + 9 × 5 = 51. The common mistake is multiplying by ten jumps because the term number is ten. Draw ten positions as ten dots and count the spaces: there are only nine gaps. This distinction connects pattern questions to the earlier guide about counting points and intervals.
Before using a difference table, check the kind of pattern
Not every pattern is governed by addition. The sequence 3, 6, 12, 24 grows by multiplication by two. Its differences are 3, 6 and 12, which do not form a simple constant row. A pupil who insists that every difference table must show a straight additive increase may miss the cleaner multiplicative rule. Likewise, an alternating sequence might interleave two separate patterns. A visual tile design can show a constant number of new pieces at each stage, but only if the stages are genuinely built in the same way.
Use a short decision order. First check whether the terms change by a fixed amount, positive or negative. Second inspect whether the gaps themselves change regularly. Third see if a consistent multiplication or division describes the steps more simply. Fourth check alternating positions or a growing figure. Fifth compare your proposed rule with every term and any story constraints. If it does not reproduce the earlier terms, it is not a valid explanation, no matter how neatly it gives the next one.
Three questions that prevent many mistakes
Which position is being requested? “Next term”, “eighth term” and “the number of tiles in Figure 8” are not identical instructions. What does each term count? A term might be the number of blocks, total edges, squares along a border or money collected after a fixed number of days. What changes from one stage to the next? If a row of squares shares sides, adding a square may add three new matchsticks, not four, because one edge is shared. Read the construction before calculating.
Now notice the common trap of drawing a correct table with the wrong subtraction direction. If the numbers decrease, as in 20, 17, 13, 8, the signed gaps are -3, -4, -5. A sequence of losses is still a change pattern; the next change may be -6, giving 2. Recording only 3, 4, 5 without negative signs can make you add instead of subtract. When unsure, write the full operation between consecutive terms, such as 17 – 20 = -3, rather than a bare magnitude.
Worked examination question: pattern with growing gaps
A club places 4 chairs in the first row, 7 in the second, 11 in the third and 16 in the fourth. If the pattern continues by increasing the number of extra chairs by one each row, how many chairs are in the sixth row? Identify the changes: +3, +4, +5. The fifth-row change is +6, so row five has 22 chairs. The sixth-row change is +7, so row six has 29 chairs. Notice why “sixth row” does not mean “add six”. The next increment belongs between rows four and five; one more increment belongs between rows five and six.
A complete check returns to the description. Row lengths are 4, 7, 11, 16, 22, 29, and the new chairs added are 3, 4, 5, 6, 7. The increments do indeed grow by one. If the question asked how many chairs are in the first six rows altogether, the answer would be the sum 4 + 7 + 11 + 16 + 22 + 29 = 89. Do not answer a total over rows question with the value in one row. Mark the target before doing arithmetic.
Worked examination question: reverse a rule
The number of stickers in successive envelopes forms the pattern 5, 9, 13, 17, and so on. One envelope contains 65 stickers. Which numbered envelope could it be? The gap is 4. From term 1, which has 5 stickers, the total increase to 65 is 65 – 5 = 60. Sixty contains fifteen jumps of four. Fifteen jumps take you from envelope 1 to envelope 16, so the answer is the sixteenth envelope. Check: 5 + (16 – 1) × 4 = 65. The hidden error is answering “fifteenth” because there are fifteen jumps.
Reverse problems are usually safer when you keep the first position visible. Let the term position be n. The same pattern can be described as 5 + (n – 1) × 4. Set this equal to 65: subtract 5 to get 60; divide by 4 to get 15; add 1 to recover n = 16. If symbolic notation feels unfamiliar, use the verbal chain instead. The algebra merely records the same steps. For a broader explanation of algebra and position rules, the Mathematics Learning Hub is the subject owner.
1. Fixed increase
Given: 4, 7, 10, 13. Track the changes: 3, 3, 3. Work it through: Next term 16; eighth term 4 + seven jumps of 3 = 25. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: If you write 4 + 8 × 3, you have counted eight jumps instead of the seven spaces between first and eighth terms. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Change the first term to 6 but keep the increase at three. The eighth term becomes 27, showing that both start and gap matter. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
2. Growing increase
Given: 2, 5, 9, 14. Track the changes: 3, 4, 5. Work it through: The likely next gap is 6, so the next term is 20. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Adding the last observed gap of five gives 19, but the instruction is to continue a growing-gap pattern rather than repeat a fixed gap. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Now extend one further term: add 7 to 20 to obtain 27. Check the complete gap row. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
3. Square-number pattern
Given: 1, 4, 9, 16. Track the changes: 3, 5, 7. Work it through: The next odd-number gap is 9, giving 25. The listed terms also match one squared, two squared, three squared and four squared. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Do not assume a constant difference; the gaps are not equal. A picture of square arrays makes the source of the extra 2, 4, 6 and 8 blocks visible. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: To find the seventh square number, use 7 × 7 = 49, then verify the incremental method if desired. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
4. Decreasing with increasing loss
Given: 20, 17, 13, 8. Track the changes: -3, -4, -5. Work it through: Subtract six next, obtaining 2. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Recording unsigned gaps can lead to an incorrect addition. Keep signs attached to the direction of change. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Move backwards from 8 to 13 by undoing the -5 change; this reverse check catches a sign error. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
5. Constant multiplication
Given: 3, 6, 12, 24. Track the changes: 3, 6, 12. Work it through: Doubling is the simplest stated rule. The next term is 48. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: The differences rise, but adding an imagined fixed gap would not match the sequence. Check multiplication early. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: If the number of tokens is being counted, 48 is sensible; if this represented whole people and halving continued, later fractions might be impossible in context. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
6. Triangular growth
Given: 1, 3, 6, 10. Track the changes: 2, 3, 4. Work it through: The next additional group is five, so 15. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Many pupils see 10 and guess 14 by repeating the last addition; the pattern calls for a larger addition. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Draw one extra row of five dots, then count all 15 dots to check the numerical rule against a visual model. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
7. Alternating tracks
Given: 2, 9, 4, 8, 6, 7. Track the changes: 7, -5, 4, -2, 1. Work it through: The odd positions go 2, 4, 6, 8 and the even positions go 9, 8, 7, 6. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: First differences look irregular because two tracks are interwoven. Separate terms by position before forcing a single-gap rule. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Write the next two terms as 8 and 6, then explain each from its own track. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
8. Rising even gaps
Given: 8, 12, 18, 26. Track the changes: 4, 6, 8. Work it through: The next gap is ten, so 36. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: The gap itself is increasing by two, not the term by two. A second difference row makes that visible. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Reverse check: 36 – 26 = 10, which continues the even-gap row. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
9. Increasing tens
Given: 5, 10, 20, 35. Track the changes: 5, 10, 15. Work it through: The next gap is 20; the next term is 55. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: A learner may guess doubling from 5 to 10 to 20 and incorrectly write 70 after 35. The earlier doubling does not hold for every step. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Try one more term: a 25 increase gives 80. State the rule before giving the result. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
10. Changing loss
Given: 50, 45, 39, 32. Track the changes: -5, -6, -7. Work it through: The next loss is eight, so 24. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: If the context is fuel consumed or stock sold, the loss magnitude rises even though the stock falls; do not reverse this relationship. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: A second stage of change gives losses 5, 6, 7, 8; continue to 15 only if the story allows that next loss. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
11. Successive sums
Given: 1, 1, 2, 3, 5. Track the changes: 0, 1, 1, 2. Work it through: Each new term can be the sum of the previous two. The next term is 8. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: A first difference table is not the clearest rule here. Testing the relation between neighbouring terms gives a simpler explanation. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Check the proposed rule against the first possible triple: 1 + 1 = 2; then 1 + 2 = 3. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
12. Shared-side squares
Given: 4, 7, 10, 13 matchsticks. Track the changes: 3, 3, 3. Work it through: A row of 4 adjacent square frames needs 13 sticks if each new square shares one side. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Do not count four new matchsticks for each extra square; shared sides are not drawn twice. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Draw a fifth square and confirm it adds three new sticks, giving 16. This is a construction fact, not just arithmetic. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
13. Border edges
Given: 4, 6, 8, 10 exposed unit edges. Track the changes: 2, 2, 2. Work it through: A single row of equal square tiles has an outer boundary that gains two unit edges per added tile. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Adding four edges for every tile ignores the shared interior edge. Counting all tile edges overstates the perimeter. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: With five tiles, exposed unit edges total 12. Check by counting top, bottom and two short ends. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
14. Missing middle term
Given: 7, __, 17, 22. Track the changes: 5, 5, 5. Work it through: The missing number is 12 if the stated rule is a constant addition of five. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: A guess based only on the two neighbouring numbers must be checked against both sides: 7 to 12 and 12 to 17 are five each. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Ask what happens if the final term is 23 instead. The fixed-gap hypothesis would no longer fit every shown term. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
15. Position formula
Given: 2, 7, 12, 17. Track the changes: 5, 5, 5. Work it through: The nth term is 2 + (n – 1) × 5, or 5n – 3. At position 20 the value is 97. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Do not treat n as the number of jumps from zero; from position one to position twenty there are nineteen transitions. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Check the formula on positions 1 and 4: it must give 2 and 17. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
16. Odd-number pattern
Given: 3, 5, 7, 9. Track the changes: 2, 2, 2. Work it through: The next term is 11 and the twentieth term is 3 + 19 × 2 = 41. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Odd numbers may be recognised by parity, but the requested position still needs accurate jump counting. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Start at 1 instead; position twenty becomes 39. This reveals why knowing the category of number is not enough. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
17. Cumulative total versus daily amount
Given: Day totals: 4, 9, 15, 22. Track the changes: 5, 6, 7. Work it through: If the daily additions rise by one, the next total is 30 after adding eight. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Do not confuse 22 (all items collected by day four) with seven (the number collected only on day four). Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: If asked for day-five collection only, answer eight. If asked for the cumulative count, answer thirty. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
18. Half each time
Given: 80, 40, 20, 10. Track the changes: -40, -20, -10. Work it through: Halving gives next 5. The first differences do not have a single constant step. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Subtracting ten again gives zero and ignores the multiplicative relationship present in every step. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Check whether the story allows halves. Five pieces of candy is possible; half a pupil in a headcount is not. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
19. Second difference
Given: 3, 8, 15, 24. Track the changes: 5, 7, 9. Work it through: The next gap is 11; the fifth term is 35. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: The second differences are 2 and 2, supporting an evenly growing first-difference row. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Write two rows of gaps and verify all four original terms before extending further. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
20. Time-cycle pattern
Given: Monday, Wednesday, Friday, Sunday. Track the changes: two days forward each time. Work it through: Two days after Sunday is Tuesday, not a second Sunday. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Day names wrap around a seven-day week. The words are categories whose positions form a cycle, not ordinary increasing numbers. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Label weekdays 1 to 7; after day seven, wrap to day one and continue the two-day shift. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
21. Too few clues
Given: 4, 7, 10. Track the changes: 3, 3. Work it through: A constant +3 rule suggests 13, but three terms alone do not prove that no other rule is possible. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: Do not pretend that a difference table certifies the unique law of nature. Examination wording or a drawing normally identifies the intended simple pattern. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: If a picture shows growing rows with additional tiles, use that construction to justify the next term rather than relying only on numerical appearance. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
22. First term not one
Given: 12, 17, 22, 27. Track the changes: 5, 5, 5. Work it through: Position 1 is already 12. Position 15 has fourteen jumps: 12 + 14 × 5 = 82. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: If you substitute fifteen for the number of jumps, you answer 87: exactly one gap too many. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Mark the first and fifteenth dots on a number-line sketch. Counting intervals repairs the off-by-one error. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
23. An unknown start
Given: x, x+4, x+8, x+12. Track the changes: 4, 4, 4. Work it through: Knowing only the step size does not determine x. An additional value, total or position is needed. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: A student may assume x=0 just because no first number is printed. That invents information. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: If term four is 25, work backwards by three jumps of four: x=13. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
24. Measured patterns
Given: Lengths 2 cm, 5 cm, 8 cm, 11 cm. Track the changes: 3 cm, 3 cm, 3 cm. Work it through: The next length is 14 cm if the same construction continues. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: A numerical pattern is not a reason to discard the measurement unit; every term represents a length. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Convert the lengths to millimetres: 20, 50, 80, 110. Their differences are 30 mm, consistently equivalent to 3 cm. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
25. Two missing values
Given: 4, __, __, 19. Track the changes: 5, 5, 5. Work it through: The missing entries are 9 and 14 in a four-position constant-gap sequence. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: There are three jumps from 4 to 19, so the total difference fifteen must be shared across three intervals, not four. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: Check each term: 4,9,14,19 gives exactly three increases of five. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
26. A visually changing row
Given: A line of circles grows 2, 4, 6, 8. Track the changes: 2, 2, 2. Work it through: If every new stage adds one circle at each end, stage five contains ten circles. Tell yourself which term is the starting point and which jumps belong between the starting term and the requested position. This keeps the representation attached to the meaning of the quantity rather than turning it into a detached arithmetic trick.
Common error and repair: The rule belongs to the construction, not to a memorised table. Explain why two circles are added. Before you accept a rule, reproduce each given term in order. If the rule produces a correct final number only by accident while contradicting one of the middle terms, reject it. A valid explanation needs to survive the visible evidence, not only the last calculation.
Transfer challenge: If an object is attached at one end only, the count would change by one; the physical construction selects the rule. Complete the challenge on paper without copying the first solution. Then try describing the pattern to a younger pupil using the words “position,” “change,” “start” and “check.” A good explanation should allow the younger pupil to rebuild your reasoning rather than imitate a number you supplied.
A difficult pattern question may be a counting question
In a matchstick arrangement, the terms often represent the number of sticks in one design, not a row of freely chosen numbers. If three adjacent squares require 10 sticks, four adjacent squares require 13. Why three extra rather than four? The new square shares one side with the previous one. There is one top edge, one bottom edge and one far end edge to add. Counting the common side twice creates a consistent-looking but wrong rule. A simple construction sketch is therefore evidence for the difference row.
A similar issue appears when counting holes, fence posts and spaces. Six equally spaced posts enclose five intervals between the first and last posts. If each interval is 2 m, the distance between the extreme posts is 10 m, not 12 m. This is not a special rule about fence posts. It is the general relation between positions and the transitions between them. In a number pattern, term 1 to term 6 also involves five transitions.
When the unknown sits in the middle
Suppose the pattern is 7, blank, blank, 22 and the question states that the same amount is added at every step. From 7 to 22 the increase is 15 across three steps, so each increase is 5. The two missing terms are 12 and 17. Check: 7 + 5 = 12, 12 + 5 = 17, 17 + 5 = 22. Do not divide fifteen by four simply because four positions are printed. This error comes from confusing terms with gaps.
Now change the instruction: the increases grow by two each time, and the row is 3, 8, blank, 24. The first increase is five. The following increases are seven and nine, so the missing value is 8 + 7 = 15. Check the last step: 15 + 9 = 24. The extra condition changes the method. Without it, several rules might be invented to fill the blank. Read constraints before calculations.
What if several rules seem possible?
Finite number lists admit many invented mathematical patterns, especially if you allow complicated expressions. PSLE questions are not normally asking you to prove that a mathematical rule is the only imaginable rule for a short sequence. They ask you to use a stated construction, a sensible progression or explicit condition. Look for wording such as “Each new figure has two more squares than the previous figure” or a table showing stages. The justified rule is the one that respects both the values and the context.
If there is no story, no explicit rule and too little evidence, avoid claiming certainty. For instance, 2, 4, 6 suggests 8 if the instruction is to continue even numbers by adding two. But someone could define a different rule matching those three numbers and giving a different fourth term. In primary-school exercises the conventional simplest progression is usually intended; still, the habit of checking the question’s actual conditions prevents overconfident extrapolation. This is a lesson about mathematical reasoning, not a reason to refuse ordinary school questions.
Independent set: calculate, explain, check
Independent item 1. 8, 13, 18, 23: sixth term. Answer and check: The gap is 5. Sixth term = 8 + 5 × 5 = 33. Five jumps, not six. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 2. 1, 5, 10, 16: next term. Answer and check: Gaps 4, 5, 6 imply the next gap 7. Next term 23. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 3. 30, 26, 21, 15: next term. Answer and check: Losses 4, 5, 6 imply another loss of 7. Next term 8. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 4. A row of one square has 4 sticks and each new square adds 3: seven squares. Answer and check: 4 + 6 × 3 = 22 sticks, because six extra squares are added to the first. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 5. 10, 20, 40: next term if it doubles. Answer and check: 80. Do not treat the last difference 20 as a permanently fixed amount. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 6. 6, 10, 14, 18: find the term equal to 70. Answer and check: (70 – 6)/4 = 16 jumps, so position 17. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 7. 4, blank, 12, 16: fixed increase. Answer and check: The gap is 4. Missing term 8. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 8. 1, 3, 6, 10: sixth term with growing increments. Answer and check: Fifth term 15 and sixth term 21. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 9. 9, 14, 21, 30: next term. Answer and check: Gaps 5, 7, 9, so next 11 and term 41. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 10. 100, 50, 25: next value if halved. Answer and check: 12.5; check whether context permits a fractional quantity. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 11. Four posts at 3 m apart: distance between first and fourth. Answer and check: Three intervals of 3 m = 9 m. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Independent item 12. 2, 9, 4, 8, 6, 7: next term. Answer and check: The odd-position values rise by two. Next is 8. After marking, put away the worked line and recreate the reason for the rule. If you can only name the answer, you have checked memory of an answer rather than control of a method. Change either the starting value, the target position or the unit and try again without looking.
Check the final answer using a different route
To check an additive pattern, work backwards from the proposed result: subtract the most recent gap, then the previous gap, until you return to a known term. To check a position formula, substitute two or three early positions, not only the requested one. To check a visual pattern, count objects or boundary edges in an actual drawing of a small stage. To check a word problem, attach units and explain what the final number means. A check that merely repeats the same arithmetic can reproduce the same mistake.
If a pupil uses a table, the columns need names. Write “stage number,” “number of pieces” and perhaps “extra pieces added.” A row labelled only “1, 2, 3, 4” makes it too easy to confuse which number counts the stage and which counts pieces. The words on the page are part of the Mathematics. If two solutions use different representations but agree, you gain more confidence than if you have performed the same addition twice.
A realistic practice ladder for Primary 6
Begin with five sequences that clearly use a constant addition or subtraction. For each, write the first differences and an inverse check. Next mix in five increasing-gap sequences and ask the learner to say what is changing about the changes. Then use visual matchstick and tile arrangements; require a verbal explanation of one new stage. After that, introduce missing terms, a far-away position and a reverse-position question. Finally, mix sequences with one multiplicative rule and one alternating rule. The learner should decide which tool fits, instead of being told in advance to use differences.
Repeat selected problems after a delay. On the next day, use different starting values and a new context rather than rereading the same solved sheet. Later, include one word problem where the answer is a total accumulated across stages instead of the value at one stage. Ask the learner to justify the difference between those targets. If errors return, identify the first failure: reading the question, term-versus-gap counting, signed differences, choosing the wrong rule, or basic arithmetic. Repair that failure specifically.
Frequently asked questions
Must every number pattern have constant differences?
No. Some have changing differences, multiplication, division, alternating rules or construction-based changes. Use a difference table to inspect a relationship, then choose the supported rule. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
Why do I subtract the first term before using a term number?
A constant-gap formula counts jumps after the first term. Position n is reached through n minus one jumps from position one. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
Can two correct-looking rules give different next terms?
Yes, short sequences do not logically force one unique mathematical rule without constraints. Use the story, figure and explicit instruction to justify the intended progression. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
What if the sequence decreases?
Record negative changes. If the losses increase, subtract the next loss; do not drop the signs. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
How do I know whether to sum the row?
Underline the request. “How many in stage six?” asks for a term; “How many altogether in stages one to six?” asks for a sum. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
What if I can see the pattern mentally?
That is useful. Write just enough steps to prove it: the proposed rule and a check using the given terms. Avoid unexplained guesses for multi-mark work. Practise a brief example independently, then alter one detail and test again. The ability to explain the change is more reliable evidence of understanding than repeating a definition.
Parents and tutors: observe the wrong step, not just the wrong number
Listen for “I just knew the next number.” Ask the learner what is unchanged and what changes at each step, then ask them to recreate an earlier term with the same rule. If they can explain the first two changes but not the term position, draw dots with spaces between them. If they misread a picture, rebuild the shape using a small set of real counters. If they only succeed on tables labelled “pattern,” mix in a word problem where the quantity is not named as a pattern at first.
Short practice should rise from demonstration to independent selection of method. Avoid converting every question into an algebra formula too early. A labelled difference row, a simple drawing and one sentence may be enough for a Primary 6 learner. Once that foundation is secure, symbolic notation can compress the same reasoning. Preserve a record of recurring errors: missing a minus sign, counting one excess jump, confusing stage count with cumulative total, or guessing from too few terms. Revisit each error with a changed example.
The next route: from a pattern to a defendable answer
For your next attempt, choose a fresh problem, write its stages and changes, describe the rule in one sentence, find the requested value and check an earlier step. If it still feels confusing, use the eduKateSG Mathematics Learning Hub for the wider number, algebra and representation routes. The PSLE Learning Guide connects this workshop to the English and Science lanes. The earlier Vol 0273 guide on points and intervals is particularly useful when you are repeatedly one position out. For reading conditional predictions in a question’s story, Vol 0292 offers a language contrast. A method you can explain, vary and check is more dependable than a guessed next number.