Pattern | table | graph | rule
Lesson overview
Outcomes
Students work through the process of, and make
connections between, creating a geometric pattern;
writing a table of values; plotting points on a graph; and
forming a rule using the Bridge builder learning object, a
SMART notebook file and plotting points in GeoGebra.
Stage 4
Software used
Learning objects, SMART Notebook, GeoGebra
PAS4.1 Uses letters to represent numbers and
translates between words and algebraic symbols.
PAS4.2 Creates, records, analyses and generalises
number patterns using words and algebraic symbols in
a variety of ways.
PAS4.5 Graphs and interprets linear relationships on
the number plane
Stage 5
PAS5.1.2 Graphs linear relationships from equations.
Stage 6 General Mathematics
AM2 Modelling linear relationships
Activity 1 Introduction
Hanging out the washing
Take a piece of rope, some pegs and some tea
towels into class. Tie the rope across the
classroom and start to hang out the tea towels –
using two pegs per tea towel. Pause and ask the
students these questions:
Take the tea towels off and start rehanging them
with one peg per towel, asking:

If I was short on pegs one day and could only
use one peg per tea towel, how many pegs
would I need to hang out 50 tea towels?

Are you sure?

How did you work it out?

How many pegs will I need to hang out 50
tea towels?

What about pegs for 100 tea towels, 67 tea
towels...?

Are you sure?

What’s the rule to help me work it out?

How did you work it out?

How many pegs will I need to hang out 100
tea towels, 34 tea towels... ?

What is the rule that will work out the number
of pegs I need for any amount of tea towels?
Ask students for other suggestions of ways to hang
tea towels and discuss the rule needed to work out
the number of pegs. A pattern students often
suggest is when two towels share a common peg.
The same problem from a different perspective
If I had a limited amount of pegs:
© Commonwealth of Australia 2009

How many tea towels could I hang out using my
two pegs per towel system if I only had 30
pegs? 100 pegs? 23 pegs?

How did you work it out? What is the rule?

What is the rule using the one peg per tea towel
system? Using the share a peg system?
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Teaching notes
Activity 2 Demonstration
The introductory activity is designed to actively
engage students and lead into algebraic thinking
of generalising a pattern into a rule.
Using learning objects
Encourage students initially to give a verbal
description of the pattern in words. For example,
you need two pegs for every tea towel.
Download and use the Bridge builder: triangles 1
learning object by searching on TaLe using the
search term Bridge builder.
Then suggest students could represent the
number of tea towels with a pronumeral ‘T’ and
the number of pegs with a pronumeral ‘P’. Ask
your students if they think T = 2P is correct. Have
them justify their answers.
The important concept to develop is the notion of
a variable. There is a relationship between T (the
number of tea towels) and P (the number of
pegs). If the number of tea towels to be hung
varies so do the number of pegs required to hang
them and vice versa.
Encourage students to record the pattern in a
vertical table. For example,
T
P
1
2
2
4
3
6
Using a vertical table enables students to clearly
see the ordered pairs of points to be plotted on
the number plane later.
Challenge task
Ask students, in pairs, to describe three different
ways of hanging out tea towels. For each
scenario they need to:




Draw the washing line showing the tea
towels and the pegs
Describe the pattern in words
Record the pattern in a vertical table
Write the rule for the pattern
© Commonwealth of Australia 2009
Demonstrate this resource to students using a laptop
and data projector or an interactive whiteboard.
Alternatively, students could access the resource
directly from TaLe using their laptops.
Work through the resource by following the
instructions embedded in the learning object. Ask
students for their input along the way by using miniwhiteboards or other methods.
Make clear verbal and visual connections between
the pattern formed, the way the table is completed
and the plotting of the points.
More learning objects
The TaLe website includes more learning objects
that support development of the concept of a
variable. Search TaLe using these search terms:









Bridge builder: triangles 2
Bridge builder: quadrilaterals
Bridge builder: complex squares
Bridge builder: complex pentagons
Circus towers: triangular towers
Circus towers: square stacks
Circus towers: square pyramids
Circus towers: rectangular prisms
Circus towers: triangular prisms
Page | 2
Activity 3 Investigation

Points can be hidden by clicking the blue dot
next to the point in the algebra window.

To delete points, right
click on the point in either
the drawing pad or
algebra window and
select delete.
Using SMART Notebook file
Open the SMART Notebook file Patterns. This file
may be used for teacher demonstration or as a
file students interact with on their laptops.
To draw a line in GeoGebra, choose the Line
between two points tool
Decide how important it is for your students to
create a physical model of each pattern. The
visual image may assist them to complete the
table.
Students complete the table; either within the
Notebook file on their laptop or in their maths
book and then complete the rule in words.
Click once on the first point, then once on the
second point. The equation of the line will appear in
the algebra window.
Drag down the pull tab at the top of the screen.
Students now open GeoGebra and plot the points
in the table (you may need to demonstrate how to
do this the first time).

Type points into the input bar at the bottom of
the window in the same way as you would
write them eg (0,6) enter

You can choose to show or hide the grid and
axis by clicking on View and selecting or
deselecting the options shown.
Note: If students are filling in answers onto their
Notebook file, you may like them to take clips of
each page when completed and put into OneNote
alongside the graph they create in GeoGebra.
Work through pages 2-7 of the Notebook file
Patterns either as a class, individually or in pairs.
Discuss page 8 as a class.

Points are listed in the
algebra window (on the left
hand side) in the order they
were plotted.

The blue pages are examples of one step rules.

The green pages are examples of two step
rules.

The green page patterns have joining sections.
Challenge the class to come up with other one or
two step rule patterns.
© Commonwealth of Australia 2009
Page | 3
Play a game of Guess my rule.
Allocate pages 9-13 to either individual or pairs of
students.
Students can:

draw up a table of values (in their books,
Excel or Word)

plot the points in GeoGebra

work out the rule.
If students find this easy, challenge them to work
out the rule without plotting points in GeoGebra
i.e. from the points in their table of values or
straight from the diagram.
Students present their findings to the class.
Students work through pages 18-20 individually.
Use page 21 and the way GeoGebra describes
the line (in equation form) to discuss the use of
symbols to represent words e.g. N represents the
number of toothpicks.
Page 21
Students work through pages 22-24 either
individually or in pairs, plotting points to check
their work.
© Commonwealth of Australia 2009
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