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Year 6 Maths Patterns & Rules

Geometric (growing) patterns

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Theory

In a growing pattern each figure adds the same number of matchsticks or tiles. A row of squares made from matchsticks adds \(3\) sticks each figure, plus \(1\) stick that closes the left end, so the rule is multiply the figure number by \(3\), then add \(1\). This Year 6 maths page shows how to find rules like this from pictures and tables, use them for any figure, and work backwards.

A figure is one picture in the pattern (some books call it a stage). Figure \(1\) is the first, figure \(2\) the next, and so on.

The number added each figure is the gap between neighbouring numbers in the table. It becomes the multiplier in the rule.

The extra number in the rule is the part of the picture that is there only once, such as the first upright stick of a row of squares or the centre tile of a cross.

A rule such as number of sticks = figure \(\times 3 + 1\) works for every figure, so it gives far-off figures quickly.

Shapes that do not share sides use all their sticks: separate triangles need \(3\) sticks each.

Matchstick squares, figures 1 to 3 A row of matchstick squares: figure 1 is one square of 4 sticks, figure 2 two squares sharing a side (7 sticks), figure 3 three squares (10 sticks). Figure 1 Figure 2 Figure 3
Squares: \(4, 7, 10, \ldots\) Each figure adds \(3\) sticks; the first upright stick is the extra \(1\).
Matchstick triangle strip, figures 1 to 4 A strip of matchstick triangles pointing up and down: figure 1 has 3 sticks, figure 2 has 5, figure 3 has 7 and figure 4 has 9. Figure 1 Figure 2 Figure 3 Figure 4
Triangle strip: \(3, 5, 7, 9, \ldots\) Each figure adds \(2\) sticks: the rule is figure \(\times 2 + 1\).
Tile crosses, figures 1 to 3 Crosses of square tiles with an orange centre tile: figure 1 has one tile on each of four arms (5 tiles), figure 2 two on each arm (9 tiles), figure 3 three on each arm (13 tiles). Figure 1 Figure 2 Figure 3
Crosses: \(5, 9, 13, \ldots\) Four arms grow by one tile each; the centre tile is the extra \(1\).

Rules for some growing patterns:

PatternFigure 1Added each figureRule
Row of squares\(4\)\(3\)figure \(\times 3 + 1\)
Triangle strip\(3\)\(2\)figure \(\times 2 + 1\)
Row of pentagons\(5\)\(4\)figure \(\times 4 + 1\)
Row of hexagons\(6\)\(5\)figure \(\times 5 + 1\)
Tile cross\(5\)\(4\)figure \(\times 4 + 1\)
Separate triangles\(3\)\(3\)figure \(\times 3\)
The gap is the multiplier. If the table goes up by 5 each figure, the rule starts with figure × 5.
Do not just multiply figure 1. Figure \(9\) of the squares is \(28\), not \(9 \times 4 = 36\).

Finding the rule

  1. Make a table of figure and number of sticks (or tiles).
  2. Find the gap: that is the multiplier.
  3. Find the extra: figure 1 minus the multiplier.
  4. Check the rule on another figure.

Working back to the figure

  1. Take away the extra number.
  2. Divide by the multiplier.
Example 1 — Squares: a near figure
How many matchsticks are in figure \(6\) of the row of squares?
Solution
\(6 \times 3 + 1\)\(=\)\(19\)
6×3+1=19

\(19\) matchsticks.

Example 2 — Squares: a far figure
How many matchsticks are in figure \(30\) of the row of squares?
Solution
\(30 \times 3 + 1\)\(=\)\(91\)
30×3+1=91

\(91\) matchsticks.

Example 3 — Work back
One figure of the row of squares uses \(46\) matchsticks. Which figure is it?
Solution
\(46 - 1\)\(=\)\(45\)
\(45 \div 3\)\(=\)\(15\)
(46−1)÷3=15

Figure \(15\). Check: \(15 \times 3 + 1 = 46\).

Example 4 — Triangle strip
Use the second diagram.
(i) How many sticks are added each figure?
(ii) How many sticks are in figure \(9\)?
(iii) How many sticks are in figure \(25\)?
Solution
\(9 \times 2 + 1\)\(=\)\(19\)
\(25 \times 2 + 1\)\(=\)\(51\)
19 and 51

(i) \(2\)
(ii) \(19\)
(iii) \(51\)

Example 5 — Hexagons
A row of hexagons shares upright sides: \(6, 11, 16, \ldots\) sticks. How many sticks does figure \(9\) need?
Solution
\(9 \times 5 + 1\)\(=\)\(46\)
9×5+1=46

\(46\) sticks. Each new hexagon adds \(5\) sticks.

Example 6 — Tile cross
Use the third diagram. How many tiles are in figure \(7\), and which tile is the "\(+1\)"?
Solution
\(7 \times 4 + 1\)\(=\)\(29\)
7×4+1=29

\(29\) tiles. The \(+1\) is the centre tile; the four arms have \(7\) tiles each.

Example 7 — Shared or separate?
Figure \(8\) is made two ways: \(8\) separate triangles, or a strip of \(8\) triangles. How many more sticks do the separate triangles need?
Solution
\(8 \times 3\)\(=\)\(24\)
\(8 \times 2 + 1\)\(=\)\(17\)
\(24 - 17\)\(=\)\(7\)
24−17=7

\(7\) more sticks, because shared sides are only counted once in the strip.

Example 8 — Find the mistake
Ruby says figure \(9\) of the row of squares uses \(36\) sticks, because figure \(1\) uses \(4\). What is right?
Solution
\(9 \times 3 + 1\)\(=\)\(28\)
9×3+1=28

Only \(3\) sticks are added each figure; the extra stick is there once. Figure \(9\) uses \(28\).

Common pitfalls

Multiplying figure 1. Figure 1 includes the extra stick, which is only there once.
Doubling a figure to get double the figure. Figure \(12\) of the squares is \(37\), not twice figure \(6\) (\(2 \times 19 = 38\)).
Counting shared sides twice. In a strip, a shared side is one stick.
Not checking the rule. Test the rule on two figures before using it.

Frequently asked questions

What is a growing pattern?

A pattern of shapes where each figure adds the same number of pieces.

How do I find the rule?

The number added each figure is the multiplier; then find the extra so that figure 1 fits.

What does the "+ 1" mean in figure × 3 + 1?

The one piece that is there only once, such as the first upright matchstick.

How do I find which figure uses 46 sticks?

Take away \(1\) and divide by \(3\): figure \(15\).

Why does a strip of triangles add only 2 sticks?

Each new triangle shares one side with the triangle before it.

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