Grade 5 Rules and Correspondence

What this problem type covers

Twenty-five problem types across five levels: direct and reverse scaling, choosing a correspondence expression, completing a proportional table, applying and reversing multiply-then-add rules, finding distant terms and positions, recovering a hidden input from number pairs, cycling numbers through labelled points, chaining two rules, joined-shape material counts, reaching a target under a constant rate, simultaneous cuts, repeated multiplication, nearest terms, minimum and below-bound thresholds, comparing two changing quantities, maximum regions and intersections, reversing an accumulated count, a square rule compared with a linear rule, ages that reach a given multiple, catching a head start, work-and-rest finishing times, geometric accumulation, and one or two layers of accumulated sums.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   The numbers in the two rows follow one rule.
   Top: 1, 2, 3, 9
   Bottom: 10, □, 30, 90
   Find the number in the box.
   • ①21
   • ②19
   • ③2
   • ④30
   • ⑤20
   Show solution

   ⑤20

   1. Multiply a top-row number by 10.
   2. 2 × 10 = 20.
2. 2 · ★2
   Write the numbers from 0 in order at points A, B, C, D, returning to A after the last point. Find the 31th number written at point B.
   • ①124
   • ②31
   • ③121
   • ④125
   • ⑤120
   Show solution

   ③121

   1. Numbers at the same point rise by 4.
   2. The first one there is 1.
   3. 1 + 4 × 30 = 121.
3. 3 · ★3
   Stage 1 has 13 pieces. Each new stage has 3 times as many. Find the number at stage 6.
   • ①3,162
   • ②234
   • ③3,159
   • ④1,053
   • ⑤3,158
   Show solution

   ③3,159

   1. There are 5 changes after stage 1.
   2. Multiply 13 by 3, 5 times.
   3. That gives 3,159.
4. 4 · ★4
   Joined tables seat 3, 10, 17, 24, … people. What is the least number of tables needed for 261 people?
   • ①39
   • ②37
   • ③38
   • ④36
   • ⑤40
   Show solution

   ③38

   1. The values start at 3 and rise by 7.
   2. Value 38 is 262; the neighbouring value is 255.
   3. The boundary is therefore at 38.
5. 5 · ★5
   At position ○, the white count is 8 more than ○ multiplied by itself, and the black count is 2 times ○. At which position is their difference 368?
   • ①18th
   • ②20th
   • ③22nd
   • ④21st
   • ⑤19th
   Show solution

   ②20th

   1. Test whole-number positions near the expected value.
   2. 20 × 20 + 8 − 20 × 2 = 368.
   3. The required position is the 20th.

Where people usually go wrong

• Using addition when the table shows a constant multiple.
• Undoing the multiplier before undoing the final addition or subtraction.
• Using the position itself as the number of jumps from the first term; there is one fewer jump.
• Counting every side of every joined shape and charging for shared sides twice.
• Multiplying by the number of pieces when a rod needs one fewer cut than pieces.
• Choosing the term below a target without checking whether the next term is closer.
• Claiming a minimum without checking that the previous value still fails the condition.
• Adding two changing quantities when the question asks for their difference.
• Resting after the final cut even though the work is already finished.
• Taking only the newest generation in a repeated pattern instead of the accumulated total.