Junior School Number Patterns: Build and Test a Rule

Using algebra to describe number patterns

A useful pattern rule tells you how to find a value at a position you have not drawn. In this junior-school activity, you will build a small display, record its counts, explain an algebraic rule and test it. You need multiplication, addition and substitution. Paper sketches are enough; no special equipment is needed.

Build the pattern before guessing a rule

Imagine a display made from square tiles and two round markers. At stage 1 there is one column of three square tiles. Each new stage adds exactly one more column of three squares. The same two round markers stay beside the display. Count every square and every round marker as one piece. The shapes identify two roles; their sizes do not affect the count.

First three stages: each square symbol is one tile; each circle symbol is one fixed marker
Part of display	Stage 1	Stage 2	Stage 3
Top row of squares	■	■ ■	■ ■ ■
Middle row of squares	■	■ ■	■ ■ ■
Bottom row of squares	■	■ ■	■ ■ ■
Fixed round markers	● ●	● ●	● ●

Text equivalent: stage 1 has three squares and two markers; stage 2 has six squares and two markers; stage 3 has nine squares and two markers. The three rows grow, but the two markers do not.

P1. Sketch stage 4. Record the stage number, square count and total piece count for stages 1–4. Explain what changes and what stays fixed. Try this before reading the completed table.

P1 worked response: keep the position separate from the quantity being counted
Stage number, n	Square tiles	Round markers	Total pieces, T
1	3	2	5
2	6	2	8
3	9	2	11
4	12	2	14

A strong sample response is: “Stage 4 has four squares in each of three rows, so there are twelve squares. The two markers are unchanged. There are fourteen pieces altogether.” Saying only “add three” describes the change, but does not yet tell someone which first value to use or how to find a distant stage directly.

Derive a rule from the arrangement

P2. Let n mean the stage number, beginning at 1. There are three rows with n squares in each, giving 3 × n squares. Add the two fixed markers. Therefore T = 3n + 2, where 3n means 3 multiplied by n. Read it as “three times the stage number, then add two”. In this display, n is a positive whole number and T is a count of pieces.

This explanation justifies the rule for every stage built in the stated way. The coefficient 3 counts the squares in each new column; the constant 2 counts the pieces that never change. Substitution checks our arithmetic: n = 1 gives 3 × 1 + 2 = 5, while n = 4 gives 3 × 4 + 2 = 14. Both agree with the drawings.

P3. Find the total at stage 12 without drawing every piece. The rule gives T = 3 × 12 + 2 = 38 pieces. A structural check gives twelve columns of three squares, or 36 squares, plus two markers. Do not answer “12 pieces”: twelve is the position, not the total.

Test a plausible wrong rule

P4. A learner proposes T = 5 + 3n: “There are five pieces at the start, and I add three each time.” Test the claim at n = 1. It gives 8, although stage 1 contains 5. The learner has added a new column before moving away from the first stage.

Starting from five pieces is valid, but stage n is reached after n − 1 additions. The repaired rule is T = 5 + 3(n − 1). For stage 12, this gives 5 + 3 × 11 = 38. Expanding gives 5 + 3n − 3 = 3n + 2, so these two rules describe the same arrangement. One disagreement is enough to reject the wrong rule.

P5. Could this display contain exactly 29 pieces? Yes: remove the two fixed markers from the count, leaving 27 squares. The 27 squares make 27 ÷ 3 = 9 columns, so it is stage 9. Exactly 30 pieces is impossible under this construction: 30 − 2 = 28 is not divisible by three. A fractional stage number would not represent a complete display.

A short list does not force a unique continuation

If you saw only 5, 8, 11, you might predict 14 next by continuing to add three. That is a reasonable conjecture, not proof of the only possible rule. Another instruction could say “add three twice, then add four”, producing 5, 8, 11, 15. Both instructions fit the first three values.

Here, the construction removes that uncertainty: every new stage must add one column of three squares and keep two markers. Testing the table helps find mistakes, while explaining the construction gives the reason the rule continues. Do not claim that checking three examples proves a rule for every imaginable pattern.

Independent pattern: counters removed from a tray

A different activity starts with 26 counters before any rounds. At the end of each round, exactly three counters have been removed, with none returned. The table records counters remaining after completed rounds.

Independent starting record
Completed rounds, n	1	2	3	4
Counters remaining, R	23	20	17	14

I1. Write a rule for R in terms of n. Explain both numbers in your rule.
I2. Find R after five rounds and after seven rounds. Check one answer by counting backwards in threes.
I3. A proposed rule is R = 23 − 3n. Use a known round to reject it, then repair it while keeping 23 as the starting term.
I4. How many complete rounds can take place? Explain why continuing the formula past that point does not describe this tray activity.

Explained independent answers

I1: R = 26 − 3n. The 26 is the count before any round; 3n is the number removed in n rounds. The output is counters remaining, not counters removed.
I2: R = 26 − 15 = 11 after five rounds; R = 26 − 21 = 5 after seven. From round 4, the remaining counts are 14, 11, 8, 5.
I3: At n = 1 the proposed rule gives 20, but the table shows 23. Repair it as R = 23 − 3(n − 1). The first recorded round is already complete.
I4: Eight complete rounds remove 24 counters and leave 2. There are not enough for a ninth removal of three. The formula would give −1 after nine rounds, which is not a possible count in this activity. Zero rounds is meaningful here and gives 26; stage zero was not part of the earlier display task.

Review your reasoning

Check whether you defined the input, counted the fixed part once, connected the changing part to multiplication, tested known and unseen inputs, and respected the physical limits. A correct number without an explanation leaves the rule untested. If you mix up the stage and the piece count, label the columns again before calculating.

Teachers can begin with sketches and spoken explanations, then introduce the letters. Accept equivalent rules when the learner explains the indexing. Ask a learner who has used n instead of n − 1 to identify the number of changes between stages 1 and 4. Use the tray task independently to check transfer, rather than copying the earlier plus sign.

For broader resources, visit the junior-school resource collection; access conditions vary by item. The difference-and-position background can be checked in OpenStax’s arithmetic-sequence explanation. All activities and numbers here are original teaching examples.