Growing patterns

In a growing pattern, look at what each new figure adds. That gives a rule, and the rule gives any figure without drawing it.

Tiles along a path, seats around joined tables, fence posts and rails: each one grows by the same amount every time. A rule tells you how many you need for any length.

A growing pattern is a set of figures where each one adds the same amount to the one before. Number the figures 1, 2, 3, … and call the figure number $n$ . In this lesson you find a rule that gives the number of sticks (or dots) in figure $n$ , so you can work out a figure you have not drawn.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Figure 10

Figure	1	2	3
Sticks	4	7	10

How many sticks does each new figure add? Where do they go?

A row of squares is made of sticks. Figure 1 (one square) has 4 sticks, Figure 2 has 7 and Figure 3 has 10. How many sticks does Figure 10 have?13303140

TryTry it on the board and compare the result with your prediction.

Grow the pattern

Move through the figures with the slider or the − and + buttons. The orange sticks are the ones each figure adds.

Try it yourselfYour edits are saved in this browser.

Go to Figure 10. How many sticks? Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

From pattern to rule

Here is how to turn the squares pattern into a rule.

Worked exampleBefore each step, say what you think comes next.

$n: 1, 2, 3, 4 \;\to\; s: 4, 7, 10, 13$ Count the sticks in the first few figures and put them in a table.
$+3 \text{ each time}$ Each new square needs 3 more sticks: a top, a bottom and a side.
$3n$ So figure $n$ has $n$ lots of 3 sticks...
$3n + 1$ ...plus the first stick on the left, which every figure has.
$s = 3n + 1$ The rule: sticks = 3 × figure number + 1.
$n = 10: \; s = 3(10) + 1 = 31$ Use the rule for a figure you have not drawn.

Figure 4: 4 squares. 3 new sticks for each square (12), plus the first stick on the left: 13.

Read Ben's working one line at a time. Is Figure 10 really 10 copies of Figure 1?

Use the table you filled in for the triangles.

The triangle pattern: Figure 1 has 3 sticks, Figure 2 has 5, Figure 3 has 7. Fill in the table.

figure n	1	2	3	4	5
sticks s	3	5	7

Ben works out Figure 10 of the squares pattern. Click the line where his reasoning goes wrong.

Write a rule for the number of sticks $s$ in figure $n$ of the triangle pattern (3, 5, 7, 9, …).

ModifyChange the problem on purpose and see what happens.

New patterns

Each task loads a new pattern. Use the board to see what each figure adds.

Need a hint?

The rule is s = 3n + 1, and s is 46.

Another hint

Take away the 1, then share by 3: backtracking.

Try it yourselfYour edits are saved in this browser.

How many sticks are in Figure 20 of the triangle pattern?

Each figure of the cross pattern adds a dot to each arm. How many dots are in Figure 15?

Each figure adds a column of 3 dots to a starting group of 2. Write a rule for the dots $s$ in figure $n$ .

Working backwards: which figure of the squares pattern has 46 sticks?

PractiseA few on your own, to check you have it.

Patterns and rules

A pattern has 5, 9, 13, 17 dots in Figures 1 to 4. Write a rule for the dots $s$ in figure $n$ .

A pattern's rule is $s = 5n + 3$ . How many are in Figure 12?

Using $s = 2n + 1$ (the triangles), which figure has 25 sticks?

Review: solve $3n + 1 = 31$ .

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Design a pattern

Design your own growing pattern of dots: $a$ dots in each new column, and $b$ dots to start with. Make Figure 10 have exactly 52 dots. Is there more than one way?

Stretch: write the rule for your pattern, and say how many dots Figure 100 has.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Figure 10 has 10 columns of a dots, plus b in the starting group: 10a + b.

Another hint

Try a = 5. How many must the starting group have for 52?

Set $a$ (dots in each new column) and $b$ (the starting group) so that Figure 10 has 52 dots. Show Figure 10.

Use the board. This ticks when you get there.

For teachers

Students find the rule for a visually growing pattern from what each figure adds, use it for a figure they have not drawn, and work backwards from a count (AC9M7A05, node 1). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can find what each figure of a growing pattern adds, and write a rule like s = 3n + 1.
I can use a rule to work out a figure I have not drawn.
I can work backwards from a count to the figure number.

Before class

Do the Run and Modify boards yourself first. The board draws what each figure adds in orange: that is the whole idea of the lesson.

The rule's two numbers have meanings students can see: the number in front of n is how many each figure adds, and the number added on is what every figure has before that (the first stick, the centre dot, the starting group). The rest of the course builds on this: in y = mx + c, m is the amount added each step and c is the starting amount.

The common mistake is scaling: Figure 10 = 10 × Figure 1. It fails because the parts are shared. Ben's working plants it after the method.

Running the lesson

Predict: take a show of hands. 40 (10 × 4) is the common wrong answer. Do not correct it: the board settles it.
Run: before moving to Figure 10, ask: what does each figure add, and why only 3? Point at the shared side.
Investigate: model the worked example live. Build the table, circle the +3s, then say 'n lots of 3, plus the 1 on the left'.
Modify: before the backwards task, ask: is 46 the figure number or the number of sticks?
Practise: ask students to check each rule on Figure 1 and Figure 2 before answering.
Make: collect different designs that give 52. Is there more than one?

When students get stuck

A student multiplies Figure 1 (Ben's mistake). Ask: Build Figure 2 from Figure 1. How many sticks did you add? Fix: The new square shares a side, so it adds 3, not 4. Count the orange sticks on the board.
A student writes s = 3n and is one short. Ask: Check Figure 1 with your rule. What is missing? Fix: Every figure has something before the repeating part (the first stick). Add it on.
A student puts the count in as n when working backwards. Ask: Which number in the question is the figure, and which is the count? Fix: Write the rule with the count in: 3n + 1 = 46, then undo + 1 and × 3.

Stretch and support

Stretch: A pattern of tables in a row: one table seats 4, each extra table seats 2 more people. Write the rule, then find how many tables seat 40.

Support: Give squared paper or matchsticks to build Figures 1 to 4. Fill a table before writing any rule.

Where it leads

Next: tables from rules (any rule, not only patterns), then plotting the table as points, which lie on a straight line. The two numbers of a pattern rule become m and c later in the course.

Where it leads in class: any repeated build (fencing, tiling, seating), and spreadsheet formulas that fill down.

What each step is for

Figure 10: Committing to Figure 10 before seeing it exposes the most common shortcut: multiplying Figure 1 instead of looking at what each figure adds.
grow the pattern: Seeing each new square in orange shows why the pattern adds 3, not 4: the new square shares a side.
from pattern to rule: The amount added each time is the number in front of n; what every figure has before that is the number added on. The rule gives any figure.
new patterns: One change at a time: a different amount added, a pattern of dots, a different starting group, and working backwards from a count to the figure.
patterns and rules: Finding a rule from a table, using a rule for a far figure and working backwards, mixed; the review is solving an equation from the Equations unit.
design a pattern: Designing a pattern to hit a target count reverses the lesson: from the rule's numbers to the picture.

Now you can

Find what each figure of a growing pattern adds, and write a rule like s = 3n + 1.
Use a rule to work out a figure I have not drawn.
Work backwards from a count to the figure number.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson2. Tables from rulesA rule turns each input x into an output y. Choose inputs, work out each output, and write them as a table of ordered pairs (x, y), input first.

Source: content/maths/linear-relations/growing-patterns/lesson.md