constructing figures.pdf geometry lesson

UCL SCRATCHMATHS CURRICULUM
MODULE 5:
Exploring
Mathematical
Relationships
TEACHER MATERIALS

Developed by the ScratchMaths team at the
UCL Knowledge Lab, London, England
Image credits (pg. 3): Top left: andreas (https://www.flickr.com/photos/andreas/2951113717/) [CC BY 2.0
(http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons  Top right: Sean Macintosh [CC BY-SA
4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons  Bottom left: Francisco Anzola
(Doha skyline in the morning) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons 
Bottom right: Wilson Hui from Calgary, Canada - Shanghai Skyline from the Bund, CC BY 2.0,
https://commons.wikimedia.org/w/index.php?curid=47927655

Investigation 1
Polygon Fireworks
Night Skyline
Investigation 2
Mathematically
Similar Rectangles
Investigation 3
Grid World: For
Exploring Similarity
Investigation 4
Exploring
Proportionality
3
MODULE 5:
EXPLORING MATHEMATICAL RELATIONSHIPS
INTRODUCTION TO MODULE 5
Module 5 is focused around exploring different types of mathematical relationships including
proportionality and ratio as well as introducing the concept of variable. This module could
potentially be linked with several different areas of the Key Stage 2 curriculum beyond
mathematics and computing such as geography as well as art and design.
GEOGRAPHY: LOCATIONAL KNOWLEDGE
The first investigation culminates with pupils
building polygon fireworks set against the
backdrop of a city skyline, which could be
adapted to link with specific places/regions that
pupils are learning about within the geography curriculum.
ART AND DESIGN: ARCHITECTURE
Pupils are also asked to build up polygon skyscrapers
which could be connected with learning about
specific architectural movements such as brutalist,
as part of the art and design curriculum.
SCRATCH COMPUTING MATHEMATICS
► Ask and answer blocks
► Join block
► … * …, … / … blocks
► <variable name> block
► Set <variable name> to …
block
► When … key pressed
► Variables
► Algorithm
► Repetition
► Expressions
► Definitions
► Sequence
► Debugging
► Logical Reasoning
► Decomposition
► Division, multiplication
► Angles
► Regular and irregular polygons
► Perimeter
► Randomness
► Coordinates
► Algebraic expressions
► Factor pairs
► Ratio and proportion
KEY VOCABULARY AND CONCEPTS COVERED BY MODULE 5

4
MAP OF MODULE 5
The red dashed line indicates the core activities which are important to complete before moving on to the next
module.
For activities which require pupils to continue with a project from a previous lesson you can alternatively use
the suggested INT or FINAL project for those pupils who do not have a project to continue with or if you wish all
pupils to begin from the same point.
Activity 1
Activity 4
Investigation 1
Polygon Fireworks,
Night Skyline
Investigation 2
Mathematically
Similar Rectangles
Investigation 3
Grid World: For
Exploring Similarity
Investigation 4
Exploring
Proportionality
Ask and
Answer
Starter project:
5-Polygon
Firework
Unplugged:
Polygon
Predictions
Naming
Values
Continue with:
5-Polygon
Firework
or start with:
5-Polygon
Firework INT1
The Sky at
Night
Continue with:
5-Polygon
Firework
or start with:
5-Polygon
Firework INT2
Activity 1 Activity 2 Activity 3 Activity 4
Sequences
of Squares
Starter project:
5-Altering
Polygons
Altering
Rectangles
Continue with:
5-Altering
Polygons
Exploring
Mathematica
l Similarity
Continue with:
5-Altering
Polygons
Unplugged:
Rectangle
Jungle
Enter the
Grid World
Starter project:
5-Grid World
Connecting
Corners
Continue with:
5-Grid World
or start with:
5-Grid World
INT1
Meet the
Magic Line
Continue with:
5-Grid World
or start with:
5-Grid World
INT2
Unplugged:
Module 5
Assessment
Using the
Grid World
Continue with:
5-Grid World
or start with:
5-Grid World
FINAL
BridgEing and
Solving
Problems
Continue with:
5-Grid World
or start with:
5-Grid World
FINAL

5
CONNECTIONS TO KS2 COMPUTING CURRICULUM
CURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS
Design, write and debug
programs that accomplish
specific goals
Solve problems by
decomposing them into
smaller parts
Use sequence and repetition
in programs
Work with variables
Using logical reasoning to
explain how some simple
algorithms work and to detect
and correct errors in
algorithms and programs
► In the first investigation pupils are required to design,
build and debug a program within Scratch which will
create a night skyline that can change based on user
input.
► Pupils are shown an example of the final night skyline
project (without seeing the scripts) and asked to
decompose the program into smaller parts, thinking
about the scripts that would need to be built to
replicate each of these parts.
► Pupils are required to consider the sequence of
blocks when using ask and answer to envisage what
the value of answer will be at different points in the
script.
► Pupils use different variables to control the number of
times their scripts are repeated. Repetition is also
used to create a series of rectangles to help explore
the concept of ratio and proportion.
► Pupils take their first step towards the use of variable
at the start of this module through using ask and
answer. Then later in the module pupils are required
to use variables to set and change the values of the
side lengths of polygons, the number of sides and
also the number of polygons to be drawn. The use of
variables enables the exploration of ratio and
proportion. The importance of naming conventions
for variables is also addressed during this module.
► Pupils are required to use logical reasoning when
envisaging the outcome of their scripts based on
different user inputs or different initial variable
values.
► Within an extension activity pupils are asked to
identify and suggest fixes for bugs in a number of
scripts which draw regular polygons.

6
MODULE 5: INVESTIGATION 1
Polygon Fireworks, Night Skyline
This investigation focuses on developing and building pupils’ understanding of variable through
the creation of polygon firework patterns. The initial investigation recalls the polygon patterns
from Year 5 placing a greater focus on the use of an unknown through “ask and answer” to vary
different attributes of the polygon patterns.
From a concept development perspective, the answer block is distanced from its companion
ask block within the script by prompting the pupil to vary aspects of the polygon. Variable
development continues as pupils realise the limitations of “ask and answer” and give a name to
a value by defining a user variable for the skyline towers.
Activity 5.1.1 – Ask and Answer
Activity 5.1.2 – Unplugged: Polygon Predictions
Activity 5.1.3 – Naming Values
Activity 5.1.4 – The Sky at Night
Scratch starter project 5-Polygon Firework
5-Polygon Firework INT1
5-Polygon Firework INT2
5-Polygon Firework FINAL
LINKS TO PRIMARY NATIONAL CURRICULUM
Mathematics
Solve problems, including missing number
problems, using multiplication and division
Use simple formulae
Pupils calculate the perimeter of rectangles
and related composite shapes
Find all factor pairs of a number
Draw 2D shapes using given dimensions and
angles
Distinguish between regular and irregular
polygons
(KS3) Work with experiments that involve
random numbers
Describe positions on the full coordinate
grid
► Pupils are required to discuss and build simple formula
which incorporates multiple variables and involves
multiplication and division to create their polygon
fireworks.
► [Extension] As an extension pupils are asked to build a
script for their sprite to calculate and say the perimeter
of the polygon it has drawn.
► Pupils are prompted to recall factor pairs of 360°.
► Pupils are required to use variables to specify the side
length and angle of a generalised polygon.
► Pupils are asked to discuss what is the same and what is
different between regular and irregular polygons.
► [Extension] Pupils build scripts that randomly position
polygon fireworks and towers of squares within a set
area.
► Pupils are required to use their knowledge of the full
coordinate grid to position their polygons and towers.

Explore how to use the ask and answer blocks to draw different types of regular polygons.
Explain what is the same and what is different between regular and irregular polygons.
7
MODULE 5 ● INVESTIGATION 1 ● ACTIVITY 5.1.1
Ask and Answer
❶ Pupils explore the ask and answer blocks:
they keep them isolated, click the ask
block and type in the answer. Where is the
text of the answer (the value) stored?
Pupils click the answer block to find out.
They also click the check box next to the
answer block to see its small monitor
window in the stage.
❷ Pupils build a script: When the Beetle is
clicked, it will ask What is your name?
When they answer and press Enter, the
Beetle will greet them by the name, using
the say block with the answer. They
explore the join word1 word2 block to
build a sentence for nicer greeting.
❸ Pupils modify the script: When the Beetle
is clicked, it asks what pen size it should
use, then sets it and draws a line, a square,
a regular polygon…
❹ Pupils modify the script: When clicked, the
Beetle asks what size the side of the
square (a polygon) should be.
continues on the next page
Define regular polygon [all sides equal, all
angles equal]. Show example of an irregular
polygon, e.g. a house shape. What is this
called? [an irregular pentagon] Use examples
asking: Is this irregular, is this regular? Support
pupils to define the regular polygon concept
by asking: What it is? and What it is not?
The Beetle turns through the exterior angle of
the polygon, it can be helpful to draw as a
diagram on the board. Note the connection
between the interior and the exterior angle
[interior angle + exterior angle = 180°]
Note that through the answer block we are
making the next step towards the concept of
a variable. If we ask for a value and type in
e.g. 55, we can then use the answer block as a
variable in an algebraic expression, see above
on the right.
ACTIVITY INSTRUCTIONS
LEARNING OBJECTIVES
MATHEMATICS CONNECTIONS
Pupils open project 5-Polygon Firework, Save as a copy (online) or Save as (offline) and rename.
The final version of this project at the end of Activity 5.1.1 will be 5-Polygon Firework INT1.

8
❺ [Extension] After drawing the polygon, the
Beetle will say what the perimeter of that
square (polygon) is.
❻ Pupils continue modifying their script: The
Beetle first asks for the pen size and sets it,
then asks what length the side of the
square (polygon) should be.
❼ The Beetle asks how many sides the
polygon should have, then draws such
polygon with a fixed side length, e.g. 30.
❽ Pupils switch the backdrop to the night
skyline and generalize the previous script:
the Beetle asks for the number of sides,
then draws many small polygons of that
type scattered around the sky – by jumping
to random positions. They may use the set
random pen shade block or other random
blocks. They may run the same script
several times – giving different answers to
the ask block.
❾ Encourage pupils to simplify their scripts
(making them more readable) by making a
new block polygon with the answer block
in it and use it in their scripts as a shortcut.
Ask pupils to draw a regular hexagon with a
perimeter of 180: What is the length of the
side? Draw a regular pentagon with a
perimeter of 95: What is the length of the
side? Draw an equilateral triangle with a
perimeter of 100. Write a simple formula
which connects perimeter of hexagon and
side length [e.g. perimeter of hexagon = 6 ×
side length]
Recall factor pairs of 360° from year 5, e.g. 90°
and 4 (square); 60° and 6 (hexagon); 72° and 5
(pentagon); 120° and 3 (equilateral triangle).
Why is 360° important? [The Beetle turns
through 360° as it moves around the polygon.]
An irregular polygon will also turn through
360° degrees. This fact is useful when solving
geometry problems where an angle is
unknown.
Another way of seeing this: For any regular
polygon the exterior angle of a regular
polygon is the same as the angle that a circle
is divided into.
INVESTIGATION 1
Activity 5.1.1
…continued

9
INVESTIGATION 1
Activity 5.1.1
The approach we apply here has been introduced in Module 1 and used throughout all
Y5 modules: Always build scripts “from inside out”, i.e. make sure you understand what
each ‘bit’ does, only then start combining them. The following picture is an example
sequence of such steps:
CONNECTIONS TO Y5 SCRATCHMATHS
❶
❽ In Module 2, Activity 2.3.4 we
applied another strategy: pupils
were provided with several new set
random … blocks, used the blocks
in their scripts, and only then
explored their definitions by
decomposing and modifying. Thus
they become familiar with the pick
random … to … block.
❸ In Module 2, Activity 2.1.1 we started using a pen tool of a sprite, with some of its
attributes, namely pen colour and pen size. We started using the following Pen blocks:
Pupils learned how to use the colour picker of the set pen color to … or alternatively to
use the set pen color to number_of_colour block. In the additional materials for Module
2 there is a poster with 40 colours and their number codes. Also there are several other
posters and sheets with challenges, one of them exploring the pen size, how to set it,
use it and change it.
Note that in Challenge 3: Explore the
pen size of the extension materials for
Module 2 pupils are encouraged to
use a pair of blocks set pen size to …
and change pen size by …, which
enables us to set a certain value and
then change it. This is exactly what we
will do later in this module with
variables.
Please note the blue numbers on the left link to the numbered steps in the activity instructions

ADDITIONAL SUPPORT
10
INVESTIGATION 1
Activity 5.1.1
If we click the ask block (1a), the sprite asks
the question (1b) and an edit line (1c)
appears in the stage. We type in our
response and press Enter or click the check
mark. The answer is then “stored” in the
answer report block (1d) and can be used in
our script(s). Click the isolated answer block
to see the value (1e).
To view the value in the monitor of answer,
we can also click the checkbox next to the
answer block in the Scripts tab (1f).
The key difference between the answer
block and the monitor is that the block can
be used in another block as its input (see 2
below) while the monitor is just visual
information for us to read.
The answer reporter block can be used as e.g.
an input for the say block, see (2a).
So when the Beetle is clicked it will ask the
question, then use the answer in the say block
to greet us, see (2b).
Explore the join block (in Operators) to join
together Hello and the value of the answer.
Note that we added a space at the end of Hello
so that the two words are separated by a space,
see (2c).
❶
❷

ADDITIONAL SUPPORT CONTINUED
INVESTIGATION 1
Activity 5.1.1
❹
❸
❺
❻
❼
8

12
INVESTIGATION 1
Activity 5.1.1
Pupils will make their own block jump to random position, thinking about appropriate
values for the pick random … to … It might be reasonable not to use numbers -240 and 240
but reduce them a bit so that the Beetle does not hit the edge when drawing a polygon.
❽

LEARNING OBJECTIVES
SOLUTION
Envisage the behaviour of a script which uses the ask and answer blocks in different ways.
Explain how the corresponding outcome drawing was changed by the answer.
Print and distribute the pupil worksheet 5.1.2 or do the activity as a class.
Ask the pupils to explain how the ask and answer blocks are being used in the scripts, what the
scripts will produce and whether the scripts can be simplified or improved.
13
Unplugged: Polygon Predictions
The Beetle asks for the pen size, selects a
random colour and draws a square using the
answer as the pen size. The pen size, however,
is unnecessarily set four times – inside the
repeat block. It only needs to be set once
before the repeat block.
The Beetle asks for the pen size and uses the
answer in the set pen size … block. The Beetle
then draws a square setting a random colour
for each side.
The Beetle sets a random pen colour then
draws a square. For each side it asks for the
pen size and uses it to draw that side.
The Beetle asks for the pen size but does not
use the answer anywhere. It draws a square
using random pen colour for each side, setting
pen size to 10 inside repeat again and again,
instead of setting it just once at the beginning.
The Beetle asks for the pen size and uses the
answer in the set pen size … block. The Beetle
then draws a square and increases the pen size
by the answer repeatedly after drawing each
side.
❶
❷
❹
❸
❺

14
INVESTIGATION 1
Activity 5.1.2
Read the scripts below. For each of them draw the picture it will create and explain in words
what each script will do in the box on the right.
NAME
WHAT TO DO
ASK ANSWER SCRIPTS
❶
❷
❹
❸
❺

15
INVESTIGATION 1
Activity 5.1.2
Do the following as a class: Each of the scripts below was intended to draw a regular polygon.
However, in each script there is a bug. Envisage the original intention, explain the bug and
suggest a fix.
In this script the answer is not used in the turn block at all. Therefore instead of drawing a
polygon of the answer sides, the Beetle draws only answer sides of an octagon of the fixed
size, see (a) below.
In this script the angle to turn by is wrong, the Beetle must turn by 360 / answer, that is
twice as much as it turns now, see (b).
In this script the question is asked four times, as the ask block is inside repeat. It means
that if we do not answer the same value each time, the Beetle will not draw a regular
polygon, see (c).
In this script two questions are asked but the answer to the first one is never used for
anything but overwritten by the second answer immediately.
EXTENSION ACTIVITY INSTRUCTIONS
ADDITIONAL SUPPORT
❶ ❷
❹
❸
❶
❷
❹
❸

Explore how to use variables within a script to store different values at the same time.
Explain why we now need variables to draw multiple regular polygons of different sizes.
16
Naming Values
❶ Pupils combine two questions in their Beetle script:
the Beetle should first ask about the side length of the
polygon to be drawn, then about the number of its
sides. However this is not possible using only the tools
we have already used. Observing the monitor of the
answer block, go through the script step by step so
that pupils discover this problem themselves.
❷ To remember the answer of the question asked, we
have to give that value a name – to store the value in
a variable. Pupils make a variable named side length.
They drag two isolated blocks in the scripts area: set
side length to … and the reporter block side length,
keep them isolated and explore, observing also the
small reporter window. They set different values to
the variable. Similarly, they create the second variable
number of sides.
❸ Pupils snap two blocks: a question What side length?
in the ask block and set side length to … the answer,
run it and explore the value of the side length variable
in its small monitor.
❹ Pupils build the whole script from step 1 again, asking
two questions and setting each variable to the
corresponding answer. Then they modify the polygon
block so that it uses these two variables instead of the
answer block.
❺ Pupils make the third variable number of polygons
and add another question in the script: How many
polygons? When clicked, the Beetle will ask three
questions and draw that many polygons of the size
and type as answered by the pupils.
Note that there is only one set variable to … block
with a drop down list of all the variables.
LEARNING OBJECTIVES
Pupils continue in their own version of project 5-Polygon Firework, or open the 5-Polygon
Firework INT1, Save as a copy (online) or Save as (offline) and rename. The final version of this
project at the end of Activity 5.1.3 will be 5-Polygon Firework INT2.
Note that the actual setting of a
variable happens only after you
run the block – by clicking or
running a script containing that
block.
You may prefer to do most of this
activity (up to point 4, including)
using the empty plain backdrop.

17
INVESTIGATION 1
Activity 5.1.3
Here is an attempt to solve the task but it does not work properly. The answer block
appears three times in the script, (1a) and (1c) refer to the second answer and (1b) refers
to the first answer. However, as soon as we answer the second question, the first value of
answer is lost and replaced by the second answer, see (1d). That is why the Beetle uses
value 8 for (1a), (1b), and (1c) and draws (1e) instead of intended (1f).
To make a variable we go to the Data group and click the Make a Variable button (see 2a).
After we type in the name of the new variable and click OK button (1b), several new
blocks appear in the Data group. In this activity we use the reporter block side length and
the set side length block.
❸
❷
❶
ADDITIONAL SUPPORT

18
INVESTIGATION 1
Activity 5.1.3
The Beetle asks two questions and
keeps the answers in variables side
length and number of sides. Both
variables are then used to draw a
polygon, number of sides is used
twice. (4b) is an alternative solution
using our own block polygon.
Variable number of polygons is used as the repeat value,
both side length and number of sides variables are used
inside the polygon block definition.
Encourage pupils to make and use the polygon block so
that the when this sprite clicked script is shorter and
more comprehensible.
Alternatively, both pen up and pen down blocks might
be moved inside the jump to random position definition.
❺
❹

Explore the following Surprising polygons:
19
INVESTIGATION 1
Activity 5.1.3
Star polygons are drawn by connecting one
vertex of a regular polygon to another (non-
adjacent one) and repeating until you return
to the start (the first one in the row above).
To demonstrate what is happening, try
walking around a five-pointed star, paying
careful attention to your turning. You will see
the four walls of the room twice, not once as
you would for a regular polygon. You have
turned a total of 360° twice, or 720°. All of
the star polygons here are found by using
multiples of 360°.
EXTENSION IDEAS
Choosing names for variables
Although pupils are encouraged – and
supported by Scratch – to give any name to
variables as they wish, a name can easily
become confusing, instead of helpful. To the
right, you see a real example from a school: a
pupil used the text of the question as the
name of a variable, the value. The confusion
may occur when the variable is then used in
other blocks, see (b) and (c).
The name of a variable should reflect what the ‘answer’ represents. In this case it could be e.g.
length or side length…

Explore how to draw towers of squares of different heights and in random positions.
bridgE to mathematical quantities and formulas to calculate side length or height of a tower.
20
The Sky at Night
❶ Pupils make their own block square using the side
length variable to draw it. They build a script: when the
Beetle is clicked, it will ask What side length? then draw
a tower of 10 small identical squares atop each other.
❷ It is not necessary to have only 10 floor towers. Pupils
make a new variable number of floors and build a more
powerful block tower which will draw a tower of
identical squares – defined by the number of floors
variable.
❸ Pupils modify their script for the Beetle to first ask for
the number of floors and save the answer in variable
number of floors. Then it will ask for the side length
and save the answer in variable side length and draw a
corresponding tower.
❹ [Extension] Pupils generalise their solution so that the
script draws a night skyline of many towers of different
numbers of floors and different side lengths. The script
will repeat the tower part, asking each time for the
input value – or, alternatively, setting them at random
with an appropriate minimum and maximum. All towers
will be scattered at random.
❺ [Extension] Pupils create a sky full of polygon fireworks,
then a skyline of towers, combining all previous steps.
Note that for firework part and for the skyline part it
might be useful to have two different jump to random
position blocks, so that the whole scene could be
created in one click.
Draw out the structure of the
towers on the white board, indicate
the starting and ending point of the
Beetle drawing it. Where do we
need to move to start the next
floor? What is the algorithm? [first
draw a square then move upwards
the side length]
Connect the Beetle output with
mathematical quantities and
formula. E.g. How tall is the tower?
Write as a simple formulae. [height
of tower = side length * number of
floors]
Pose questions: If a tower is 120
tall, and side length of the square is
15, how many floors does it have?
LEARNING OBJECTIVES
Pupils continue in their own version of project 5-Polygon Firework, or open the 5-Polygon
Firework INT2, Save as a copy (online) or Save as (offline) and rename. The final version of this
project at the end of Activity 5.1.4 will be 5-Polygon Firework FINAL.

21
INVESTIGATION 1
Activity 5.1.4
In Module 2, Activity 2.2.1 pupils drew a square and a equilateral triangle. In Activity 2.2.3
they were encouraged to give a name to their square script, making their own square
block. In Activity 2.2.4 they made another new block – triangle and were asked to use
these new blocks to draw a tower of two squares and also a house.
❶
In the additional support of that activity we suggested to encourage pupils to build a script
and run it step by step thinking about the questions below:
■ Where will my Beetle finish after
drawing the first square?
■ Which direction will it point in?
■ Where exactly do I want it to draw
the second square? Which block
will make the Beetle get there?
■ Will it then point in the correct
direction?
■ Where will it finish after drawing
the second square?
Now in our new square block from Activity 5.1.4 we make use of a variable side length which
is set by using the ask and answer blocks. Some pupils may come with the following solution
based on a generalization of the Y5 task:
While it looks like a correct solution for the situation when we set the value of side length to
10 (example (a) above), it is easy to demonstrate the problem if we set the value of side
length to be e.g. 20 (example (b) above). The Beetle needs move exactly by side length from
one square to the next one, whatever value it is.

22
INVESTIGATION 1
Activity 5.1.4
In the definition of the square block we use the side length variable, the value of which will
be set in the when this sprite clicked script by ask and answer.
In the script for the Beetle each square can have the same colour or may have different pen
shades or different pen colours, pupils can choose.
Pupils should start using the vocabulary: The sprite asks
for … then saves or keeps the answer (or answered value)
in a variable …
❶
❷
❹
❸
ADDITIONAL SUPPORT

23
INVESTIGATION 1
Activity 5.1.4
Alternative solution with number of floors and side length set at random, without any
asking: Carefully choose an appropriate minimum and maximum for each value, including the
ranges for x position and y position for random jumping.
❺

24
Mathematically Similar Rectangles
This investigation explores the mathematical property of similarity. The materials deliberately
use the words “mathematically similar” to draw attention to the special mathematical
properties of the term similar: having corresponding side lengths proportional (in the same
ratio) and corresponding angles equal.
Pupils start by exploring and building sequences of growing squares which use one variable and
the change <variable> by block to change the value of the variable. They develop their script to
explore and build patterns of sequences of growing rectangles which require two variables. The
idea of a “magic line” – which connects each top right corner and bottom left corner of the
sequences - is introduced as a visual tool to test if the sequence of rectangles are
mathematically similar. This idea is developed when the Beetle is programmed to say the result
(ratio) of height/base. Pupils recognize that when the Beetle says the same number, the
sequences of rectangles are mathematically similar, e.g. the rectangles are proportional.
Activity 5.2.1 –Sequence of Squares
Activity 5.2.2 – Altering Rectangles
Activity 5.2.3 – Exploring Mathematical Similarity
[Extension] Activity 5.2.4 – Unplugged: Rectangle Jumble
Scratch starter project 5-Altering Polygons
Mathematics
Know that rectangles are not always similar
to each other.
Solves problems involving similar shapes
where the scale factor is known or can be
found.
Pupils recognise in contexts when the
relations between quantities are the same
ratio (for example similar shapes).
Pupils should consolidate their
understanding of ratio when comparing
quantities, sizes and scale drawings by
solving a variety of problems.
► Pupils are required to identify from scale drawings as
well as from scripts (in Scratch) rectangles which are
proportional to one another and to explain their
choices.
► Pupils are asked to identify which sequences of squares
and rectangles are proportionally similar and to use the
idea of a “magic” line which can be drawn through the
upper right corners to help with this.
► Pupils are required to explore ratio and proportion in a
number of ways, through exploring drawing rectangles
of different ratios within Scratch using variables,
through identifying and explaining proportional scale
drawings and also through envisaging the outcomes of
existing scripts.

Explore repeatedly changing a variable by a set amount to create a sequence of increasing
squares.
Explain the difference between the change <variable> by and set <variable> to blocks.
25
Sequence of Squares
Ask pupils: Describe the side
lengths of each of your squares.
What is the final side length
drawn? What is the value of the
side length variable at the end,
explain this difference? [Starting
with 20, repeat 5 and change side
length by 20, will draw 5 squares,
20, 40, 60, 80, 100. However, the
side length variable will report
120 at the end due to the
position of the change side length
block. Write the value of the side
length variable on the white
board as you step through the
code block by block.] See
additional support ❺ for more
examples.
❶ Pupils will use the square block – identical to the one
from the previous investigation. Using this block the
Beetle should draw a square the side of which is
defined by the side length variable. Pupils build the
following short script and use it repeatedly to draw
several squares with different values of the side
length.
❷ Pupils clear the stage and use the same script to draw
a square of side 20, then 40, then 60, 80, 100…
❸ Pupils drag in the change side length by … block and
keep it isolated. They explore the difference between
set side length to … and change side length by …
❹ Pupils clear the stage and build a script to draw the
whole pattern of growing squares in one go, using the
change side length by … block.
❺ Pupils experiment with change side length by … inside
the repeat block to make the same script short.
❻ Pupils build a script to draw a row of several attached
squares: (a) of the same side, then (b) of growing side,
the first one 20 and then increasing by 10.
❼ [Extension] Pupils modify the previous script (b) so
that the squares have a gap of 10 between them.
❽ [Extension] Pupils modify the previous script so that
each square has sides between 30 and 60, chosen at
random.
LEARNING OBJECTIVES
Pupils open project 5-Altering Polygons, Save as a copy (online) or Save as (offline) and
rename.

26
INVESTIGATION 2
Activity 5.2.1
Within the 5.1.1 Connections to Y5 ScratchMaths it described using a pair of blocks to set
pen size to … and change pen size by … within e.g. Module 2: Challenge 3.
❸
For example in Module 3, Activity 3.1.3 we used
change y by … block to make Tera jump high then
slowly float back:
Note however there are other pairs of blocks working in a similar way: the first one sets a
certain value (of the x position, y position, or the pen shade etc.) and the other one
changes that value by a certain amount.
Now we are introducing another similar pair – a block to set a value of a variable and a
block to change that value by a certain increment (a positive number) or a decrement (a
negative number).

27
INVESTIGATION 2
Activity 5.2.1
Click script (a) again and
again. Observe the value of
side length. Then click script
(b) several times and observe.
ADDITIONAL SUPPORT
❶
❷
❸
❹ ❺
Below see alternative solution
of the same task. Discuss it
with the pupils.

28
INVESTIGATION 2
Activity 5.2.1
Discuss as a class, envisage the different or identical outputs of the following scripts, make sure pupils
understand these subtle differences. How many squares will be drawn? What will be the side length of
the first drawn square in each script? Of the second? Of the third? … Of the biggest one?
What is the initial value of the side length variable in each script and what is its final value after finishing
the repeat loop?
Two alternative solutions of 6 (b): the first one makes the Beetle “move” to the right, the second one
makes the Beetle “jump” to the right. Both are correct but note that the second solution will not work
properly if the Beetle does not point in direction 0 (up) at the beginning.
❻ Two alternative
solutions of 6 (a).

29
INVESTIGATION 2
Activity 5.2.1
❼
❽
Three slightly different alternative solutions, all are correct. Which one do you find easiest to read?

30
INVESTIGATION 2
Activity 5.2.1
The ideas built earlier in this activity might be used and further developed in many directions. See
some of them bellow and ask pupils to build scripts to draw these or similar outcomes.
The Beetle draws several squares of the growing size. Pupils can
experiment with different angles in the turn block to work out
good outcome.
EXTENSION IDEAS

ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS
31
Altering Rectangles
❶ Pupils build a script to draw a rectangle, e.g. of
the height 100 and base 30. They will then make a new
block rectangle.
❷ Pupils make a new variable height and use it in the
definition of the rectangle block. They draw different
rectangles by setting the value of height and running
the rectangle block.
❸ Pupils use change height by … block inside repeat to
draw various patterns of growing rectangles.
❹ Pupils make another variable – base and generalise
the rectangle script so that it draws a rectangle
specified by two variables: height and base. They set
both of them and draw different rectangles. They
experiment with the Beetle pointing in different
directions and drawing the same rectangle.
❺ There are four scripts and four outcomes shown in the
presentation. Pupils will read the scripts, envisage and
match each of them against one of the possible
outcomes.
❻ Discuss as a class what happens if we connect all
upper right corners in a pattern of rectangles by a line.
LEARNING OBJECTIVES
Explore how to draw a sequence of rectangles of increasing size using two variables.
Envisage the outcomes of scripts changing the height and/or base by different amounts.
Explain what the magic line is.
Pupils continue in their own version of project 5-Altering Polygons, or start again with the initial
starter project 5-Altering Polygons.
Connect the structure of the
above script with the symmetry
of the rectangle. [A rectangle has
two lines of symmetry, it has two
pairs of opposite and equal sides]
Pupils should create two
sequences of rectangles on their
screen where the height and the
base change. Use the edge of a
piece of paper overlaid on the
screen to support visualizing a
line connecting the upper right
corners of the rectangles. Ask
pupils to compare their lines.
What is the same and what is
different?
Encourage pupils to compare the
shape of the initial rectangle, to
the final rectangle in each
sequence. Are they the same
shape? [When the line is “magic”
the rectangles are mathematically
similar.]

32
INVESTIGATION 2
Activity 5.2.2
Encourage pupils to spot the regularity in drawing the rectangle and ensure the script
reflects that regularity.
In this step the definition of the rectangle block will
be generalised to draw rectangles of a fixed base of
25 but of a height specified by the variable height.
ADDITIONAL SUPPORT
❶
❷
❸

33
INVESTIGATION 2
Activity 5.2.2
❹
❺

34
INVESTIGATION 2
Activity 5.2.2
❻
When the line connecting
the top right corner of each
rectangle goes through the
bottom left corner it is
“magic”. The magic line
indicates that the sequence
of rectangles are
mathematically similar.
Corresponding sides are in
the same ratio, and
corresponding angles are
equal. We can say that the
rectangles are proportional
to each another.
Another connection is to use the language of enlargement: When the line Is magic, one
rectangle is an enlargement of the other. It’s important to notice that when the rectangles are
an enlargement or mathematically similar, their shape is maintained. Encourage pupils to
compare the first rectangle with the final rectangle in each sequence.
In this sequence, there is no magic line since it does not go through the
bottom left corner. If we describe the first rectangle we might say that it is
tall and thin, whereas the final rectangle looks like a square, the shape has
changed. The rectangles in this sequence are not mathematically similar,
they are not in proportion to one another.
Pupils may begin to notice and define the relationship between the
sides of each rectangle. In this example the base of the first rectangle
is twice the height, in each rectangle this relationship is maintained.
These rectangles are mathematically similar, the ratio of the side
lengths are equal.
All squares are mathematically similar since the base is equal
to the height, each square in the sequence is an enlargement
of another square.
In the next activity pupils explore calculating height / base as a
numerical result to support what can be seen visually. They
begin to use this relationship in problem solving situations.
A slide from the pupil presentation for activity 5.2.2.

35
INVESTIGATION 2
Activity 5.2.2
Challenge pupils to create interesting outcomes using the rectangle block inside a repeat.
EXTENSION IDEAS

ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS
36
Exploring Mathematical Similarity
In Activity 5.1.1 pupils used the say block to show a value
(an answer to a question, then also in the extension to
show the perimeter of a polygon). Now we will use say to
calculate and show the value of height divided by base.
❶ Pupils build a script that will set the variables height
and base to 60 and 30, then draw a corresponding
rectangle. The Beetle will then use the say … block to
calculate and show the value of height divided by
base.
❷ Pupils now attach two blocks setting the initial values
of the height and base to the setup script.
Then they build a script which draws a rectangle
(without setting first values of height and base, as
there are being set in the setup script now), says the
value of height divided by base and finally changes
height by a number and base by a number. They run
the script several times and observe the resulting
sequence of values and pattern of increasing
rectangles.
❸ Distribute the printed pupil worksheets for this
activity (2 pages) and let them record the sequence of
values for scripts in tables A to D (first page). Discuss
them as a class.
Let pupils choose the initial numbers and the numbers
to change height by and base by in the scripts, tables
E to H (second page).
LEARNING OBJECTIVES
Explore how to display the value of the height divided by the base of a rectangle.
Explain why for certain sequences of rectangles the value of height/base stays the same.
Pupils continue in their own version of project 5-Altering Polygons, or start again with the initial
starter project 5-Altering Polygons.
In the above example, What does
the 2 represent? [The length of
the height is 2 times the length of
the base]. Demonstrate a
calculation for a different starting
height and base, appropriate to
the class.
If the height and base were
swapped, what would the Beetle
say? [0.5 since the height is 0.5 of
or half of the base].
Explore what the Beetle will say
for squares and explain the result.
[The Beetle will always say 1 for
squares since squares have equal
length sides, and anything divided
by itself is 1].
When there is a “magic” line,
what will the Beetle say? [The
Beetle will repeat the same
number for each rectangle drawn,
since the base and height are in
the same proportion. The
relationship of the height to the
base is the same, the sides are in
the same ratio. The rectangles
are mathematically similar].

37
INVESTIGATION 2
Activity 5.2.3
Pupils build the say block step by step, from inside out,
trying the parts by clicking them in the scripts area. Note
that if the say block (without any for … secs) is used, to
get rid of the bubble in the stage we either use say again
or click the red Stop sign.
It may happen that some pupils get
the following outputs in the say
block.
NaN means Not a Number. The set
blocks might have been used with
“nothing” or with a text instead of a
number.
The Beetle will say Infinity if the
divisor equals 0.
Clicking the right-
hand script again
and again is similar
to running it in a
repeat loop. If
repeated manually,
pupils have time to
write down the
value of division.
Note that our
rectangle block
works even with
one or both
negative inputs. Go
through its
definition with
pupils step by step
to see why.
ADDITIONAL SUPPORT
❶
❷

38
INVESTIGATION 2
Activity 5.2.3
The purpose of the second
activity is for pupils to find
values which produce a
“magic” line e.g. the sequence
of rectangles are
mathematically similar.
Provide a more structured
task if needed. E.g. start with
writing on the board a height
of 90, a base of 30 and change
height by 10. Pupils find the
value of change base by to
keep the Beetle saying 3.
Another approach is to provide values to set starting heights and bases
as in Table 1. Encourage pupils to discover the relationship between the
values for mathematically similar rectangles. [The value of height/base
must be the same as the change height by/change base by. We can also
say that the ratio of the height to the base is the same as the ratio of the
change height by to the change base by. Adding on quantities in the
same ratio will create a mathematically similar rectangle.]
Play a ‘beat the teacher’ game. Ask pupils to choose values for base and
for height [multiples of 10 to start with] and you will calculate the values
for the change height by and change base by to produce mathematically
similar rectangles. Try the values in the script to ensure that the Beetle
says the same number and there is a magic line.
❸
h b
90 30
100 20
95 45
120 30
60 60
25 100
15 120
Worksheet Solutions
Discuss the sequences of numbers generated by the Beetle. A good starting question might be,
What is the same and what is different? What will happen to the numbers if we keep increasing
the pattern?
Sequence A – The value of height/base is decreasing. The shape of the rectangle looks more
and more square like as we continue the pattern. We can envisage that the sequence would
get very close to 1. It can’t be exactly 1 since the height will always be 20 more than the base.
Sequence B – The value of height/base is increasing. The pattern of numbers will continue to
alternate. E.g., 2, 2.5, 3, 3.5…
Sequence C – The value of height/base is decreasing. The shape of the rectangle is becoming
very long and thin since the height is always 40. We can envisage that the sequence would get
close to 0, but not exactly, since 40 divided by a large number is almost 0.
Sequence D – The value of height/base is always 2. There is a “magic” line. The starting height
is twice the base, the change in height is twice the change in base. Adding on values in the
same ratio will always create another mathematically similar (proportional) rectangle.

39
For each table A to D always build both scripts. Be sure you first set height to 40 and base to 20.
Click the second script to see the outcome and note down the value shown, again and again.
1 2
2
3
4
5
6
7
8
1 2
2
3
4
5
6
7
8
1 2
2
3
4
5
6
7
8
1 2
2
3
4
5
6
7
8
INVESTIGATION 2
Activity 5.2.3
NAME
WHAT TO DO
SEQUENCE OF RECTANGLES A SEQUENCE OF RECTANGLES B
SEQUENCE OF RECTANGLES C SEQUENCE OF RECTANGLES D

40
Model an example on the whiteboard before distributing the worksheet. Use height 60, base 30
and change height and base by 10. It is important that pupils compare their results with their
classmates to ensure that errors are spotted. The Beetle should say 2 as the first value in each of
the sequences A-D. Sequence D will produce rectangles with a magic line, rectangles that are
mathematically similar or in proportion to one another.
1 2
2 1.67
3 1.50
4 1.40
5 1.33
6 1.29
7 1.25
8 1.22
1 2
2 2.50
3 3
4 3.50
5 4
6 4.50
7 5
8 5.50
1 2
2 1.33
3 1
4 0.80
5 0.67
6 0.57
7 0.50
8 0.44
INVESTIGATION 2
Activity 5.2.3
1 2
2 2
3 2
4 2
5 2
6 2
7 2
8 2
WORKSHEET SOLUTIONS
SEQUENCE OF RECTANGLES A SEQUENCE OF RECTANGLES B
SEQUENCE OF RECTANGLES C SEQUENCE OF RECTANGLES D

SEQUENCE OF RECTANGLES G SEQUENCE OF RECTANGLES H
SEQUENCE OF RECTANGLES E SEQUENCE OF RECTANGLES F
41
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
INVESTIGATION 2
Activity 5.2.3
NAME
WHAT TO DO
For each table E to H always use both scripts, decide for the initial values of height and base,
then choose the values to change by and write in the empty holes. Build in Scratch, set the
initial values, run the second script repeatedly and note down the value displayed each time.

42
[Extension] Unplugged: Rectangle Jumble
Activity 5.2.4 consists of three worksheets. Print and distribute them or work as a class.
❶ In the first one pupils are asked to identify possible plans for the school swimming pool 60
metres by 15 metres.
❷ In the second worksheet pupils are asked to sort a jumble of rectangles into three different
groups of those having the height either 2 times the base, or 3 times, or 4 times.
❸ Pupils are asked to sort 9 scripts into three groups: scripts in each group draw proportional
rectangles. Look for the relationship within a rectangle and also between two rectangles.
All rectangles proportional to
the swimming pool 60 metres
by 15 metres are highlighted.
There are three groups of
proportional rectangles
For the third
worksheet
the solution
is
1 – 4,
2 – 5,
3 – 1,
4 – 2,
5 – 3.c
LEARNING OBJECTIVES
WORKSHEETS SOLUTIONS
Envisage which rectangles are proportional to one another.
Explain why two rectangles are proportional. bridgE to knowledge of ratio and proportion.
❶
❷ ❸

43
INVESTIGATION 2
Ext. Activity 5.2.4
NAME
WHAT TO DO
A school has a swimming pool 60 metres by 15 metres. Which of these plans could be a scale
drawing of the pool? Mark the ones you think are correct.
SWIMMING POOL PLANS
❶

RECTANGLE JUMBLES
44
INVESTIGATION 2
Ext. Activity 5.2.4
NAME
WHAT TO DO
In the top jumble all the rectangles drawn by the Beetle are proportional, the height is always 3
times the base.
In the bottom jumble there are three types of rectangles – the height is either 2 times the base,
or 3 times the base, or 4 times the base. Sort the rectangles in the bottom jumble into three
different groups, writing either 2 to 1, or 3 to 1, or 4 to 1 on each.
❷
Sort the rectangles into three groups (2 to 1, 3 to 1 and 4 to 1) the jumble below.

45
Read the scripts below. For each script in the left column find one in the right column which
would draw a rectangle that is proportional to it.
INVESTIGATION 2
Ext. Activity 5.2.4
NAME
WHAT TO DO
MATCH SCRIPTS BETWEEN COLUMNS
❸
❶
❷
❹
❸
❺

46
Grid World: For Exploring Similarity
This investigation brings together ideas from the previous two investigations: constructing
different sized rectangles using two user defined variables and the concept of a “magic line”
indicator when rectangles (or quantities) are in proportion.
Pupils code a tool which will allows them to explore proportional relationships and to solve
proportional problems. Using four arrow keys pupils move the dot to the top right corner of a
rectangle and click the Beetle. The Beetle then constructs the rectangle and the magic line
from the ‘origin’ through the diagonally opposite corner of the rectangle and displays the ratio
of one quantity (side A) to the other (side B). The process can be repeated in order to make
comparisons.
Activity 5.3.1 – Enter The Grid World
[Extension] Activity 5.3.2 – Connecting Corners
[Extension] Activity 5.3.3 – Meet the Magic Line
[Extension] Activity 5.3.4 – Unplugged: Module 5 Assessment
Scratch starter project 5-Grid World
5-Grid World INT1
5-Grid World INT2
Mathematics
Solves problems involving similar shapes
where the scale factor is known or can be
found.
Pupils recognise in contexts when the
relations between quantities are the same
ratio (for example similar shapes).
Pupils should consolidate their
understanding of ratio when comparing
quantities, sizes and scale drawings by
solving a variety of problems.
Describe positions on the full coordinate
grid; Use simple formula
► Pupils use Scratch to display the position of the beetle
on the screen using coordinates. Pupils use scale to
transform coordinates for use with different grid tile
sizes.
► Pupils are required to build a script to calculate the ratio
of one quantity (side) to the another. The Grid World
provides the opportunity for them to explore this
relationship through experimenting with different
quantities and comparing.
► Building the Grid World provides opportunities for
pupils to explore and use coordinates in context.

Envisage the size of a grid tile using coordinates.
Explore how to move horizontally and vertically by a specified number of “tiles”.
47
Enter The Grid World
In the Grid World project the Beetle
always turns left 90 when drawing a
rectangle. It starts from the bottom
left corner of the screen and points
right.
The Scratch stage uses a coordinate
grid -240 to 240 pixels horizontally
and -180 to 180 pixels vertically. 1
Beetle step corresponds to 1 pixel.
https://wiki.scratch.mit.edu/wiki/Pixel
s
Envisage a script to draw a rectangle
which is 4 steps wide and 3 steps high
using a pen size of 1. What exactly
would be drawn on the screen? How
big would it be? Explain your answer.
[The rectangle would be drawn very
small compared to the beetle:
The Scratch stage is a rectangle 480 by
360 steps, so a 4 by 3 rectangle is very
small in comparison.]
The project uses backgrounds of
different grid tile sizes to scale the
output on the stage. E.g. Using
grid=50, for each grid tile the Beetle
moves 50 steps. How many steps
would the Beetle move if it moved 4,
50 grid tiles? [The Beetle would move
200 steps.]
❶ Pupils read the setup script and run it. In this activity
the Beetle will draw rectangles, moving only along the
grid lines. By reading the mouse coordinates in the
stage pupils find out how big a grid tile is i.e. 50.
❷ Pupils verify their conjecture by dragging in two
isolated blocks – i.e. move 50 steps and turn left 90.
By clicking them repeatedly in the correct order they
make the Beetle draw a rectangle, e.g. 4 tiles wide and
3 tiles high. The Beetle should end in the same
position, pointing in the same direction as before.
❸ Pupils make their own block move 1 tile which will
make the Beetle move forward by 1 tile – either in one
‘jump’ or more slowly and smoothly.
❹ Rectangles to be drawn will be set by two variables – A
to define how many tiles it will have horizontally and B
to define how many tiles it will have vertically. Pupils
make these variables and build a script: when the
Beetle is clicked, A and B will be set to certain values
(e.g. 4 and 2) and the Beetle will draw the rectangle.
They may make their script short and more readable
by making two more new blocks – move horizontally
(by A tiles) and move vertically (by B tiles).
❺ Pupils explore other backdrops, finding out what their
names are and what their grid tile sizes are. They
change backdrop to grid 20 and modify their scripts so
that the Beetle draws rectangle A by B tiles correctly.
❻ Pupils make a new variable grid, set it to 20 and use it
in the move 1 tile definition.
❼ [Extension] Pupils experiment with their move 1 tile
definition to make the Beetle move quickly or slowly…
LEARNING OBJECTIVES
Open the project 5-Grid World FINAL within Scratch on the IWB to demonstrate how the final
exploration tool works.
Pupils then open project 5-Grid World, Save as a copy (online) or Save as (offline) and rename.
The final version of this project at the end of Activity 5.3.1 will be 5-Grid World INT1.

48
INVESTIGATION 3
Activity 5.3.1
The actual coordinates of the mouse cursor within the stage
are displayed under the lower right corner of the stage.
The Beetle may move by 1 tile either in one go
(one move) or by repeating smaller step
several times.
Note that if the Beetle is clicked again
while still drawing the rectangle, two
drawings will somehow run in parallel
and both might be spoilt.
ADDITIONAL SUPPORT
❶
❷
❸
❹

49
INVESTIGATION 3
Activity 5.3.1
The name of the second backdrop is grid 20 and, naturally, the size of the grid tile is 20 by
20. The only detail in the script which must be modified so that the Beetle again draws
rectangles set by the numbers of the tiles A and B, is the move 1 tile definition. On the
right there are two alternatives for how to do it.
It might be helpful to have three small scripts to switch
between different grids easily. All other scripts will
work properly, if we use the grid variable to define the
move 1 tile block.
Here are different definitions of
the move 1 tile block. Envisage
which of them will make the
Beetle move very quick, very
slow, slow…
❺
❻
❼
EXTENSION

Explore how to draw a rectangle that is defined by the position of the dot sprite.
Explain how to control the sprite using the arrow keys.
50
[Extension] Connecting Corners
Be careful to set the initial
position of the Dot in its setup
script to match the initial values
of A and B. If grid is 20 and we set
A to 3 and B to 2, then the initial
position of the Dot must be 3 ×
20 to the right and 2 × 20
upwards from the Beetle’s initial
position.
Pupils will build a new way to set the sides of the
rectangle: the Dot sprite will react to pressing the arrow
keys. It will move up, down, left or right, always from one
grid point to another. These movements will increase or
decrease variables A and B by 1. Then, when the Beetle is
clicked it will draw a rectangle with the Dot sitting in its
diagonally opposite corner.
❶ Pupils modify the setup script of the Dot: they delete
the hide block and make the Dot visible.
❷ As the Dot will now be responsible for setting and
changing variables A and B, pupils move the set A …
and set B … blocks from the Beetle’s when this sprite
clicked script into the setup script of the Dot and set
them to 3 and 2.
❸ Pupils build the when right arrow key pressed script
for the Dot: the sprite will point to the right, move 1
tile in that direction and change the value of A by 1.
❹ In a similar way pupils build three more scripts for the
Dot to react correspondingly to the other three arrow
keys.
❺ Build one more script for the Dot: it will forever say the
actual values of A and B, e.g. 3, 2 or 9, 1…
❻ Pupils switch the backdrop to grid 10 and modify all
necessary bits of scripts so that the whole project
works correctly again.
❼ [Additional Extension] Pupils try to fully automatize
switching from one grid to another. What changes?
❽ [Additional Extension] Pupils modify the when …
arrow key pressed scripts so that when the Dot
reaches the grid dots closest to the edge of the stage, it
would not try to move any further.
LEARNING OBJECTIVES
Pupils continue in their own version of project 5-Grid World, or open the 5-Grid World INT1,
Save as a copy (online) or Save as (offline) and rename. The final version of this project at the
end of Activity 5.3.2 will be 5-Grid World INT2.

51
INVESTIGATION 3
Ext. Activity 5.3.2
We can check what is the initial
position of the Beetle and for the
Dot increase those x and y positions
by 3 × grid tile size and 2 × grid tile
size respectively.
Note also that if we delete the hide
block and click the show block
once, we in fact do not need it in
the setup script at all.
The Beetle is going to use
variables to draw a
rectangle A by B grid tiles
but not setting (initializing)
them nor changing them.
Therefore we must delete
those lines from the script.
We can copy the move 1 tile
block from the Beetle as the Dot
may move in exactly the same
way.
Moving the Dot sprite to the right
from the Beetle means increasing
variable A by 1.
Discuss with your class which arrows affect variable A and how and which arrows affect
variable B.
ADDITIONAL SUPPORT
❸
❷
❶
❹
❺

52
INVESTIGATION 3
Ext. Activity 5.3.2
Whenever the backdrop is switched, the grid size, i.e. the grid variable changes its value.
What else should happen as an automatic consequence of that change?
• The backdrop has switched, but there might be some lines drawn from our previous
explorations, so it should be cleared.
• Both sprites should be initialized by running their initial scripts.
Both issues above can easily be solved by broadcasting a message after resetting the grid
variable. That broadcast will run the same setup scripts as clicking the green flag.
Note that the initial position of the Beetle is the same for grid 50,
grid 20 or grid 10… (the backdrops were designed this way).
However, the Dot should be moved to a position derived from the
actual value of the grid variable – for example, always setting
initial values of A to 3 and of B to 2 and then moving the Dot A
tiles to the right from the Beetle and B tiles upwards.
Note that if we later decide to use different initial values for A and
B than 3 and 2, we simply modify the set blocks in this script only.
In fact, the only detail to modify is the initial position of
the Dot. So we can easily find out correct new
coordinates by adding 3 × 10 and 2 × 10 respectively.
ADDITIONAL EXTENSION SUPPORT
❻
❼

53
INVESTIGATION 3
Ext. Activity 5.3.2
Let us analyse what should happen when right arrow
key pressed (the scripts for other three arrow keys
would need similar analysis and modifications).
When do we want the Dot to react to the right arrow
pressed? Only in the case when certain condition is true
– in general, only if the Dot is still not “too far to the
right”. So the general structure of the script might be:
If the grid size is 50, the condition could be as simple as (a) above: if the Dot has
already moved 8 grid tiles away from the Beetle (in the horizontal direction), it should
not move any further.
However, if the grid size changes, this number will be different. For example, with the
grid 20 it should be 21, see condition (b).
The general solution (in the sense of advanced extension 7) could e.g. check whether
the Dot is still at least a grid size away from the edge, see (c) above.
ADDITIONAL EXTENSION SUPPORT CONTINUED
❽

Explore how to draw the “magic line”.
54
[Extension] Meet the Magic Line
The blue and red pen coloured
lines are a useful visual aid for the
relationship between the
coordinates of the top right
corner and the side lengths of the
rectangle. The length of the red
line corresponds to the horizontal
distance in the coordinate, and
the blue line the vertical distance.
If the dot is at coordinate (12,8),
the Beetle will draw from the
origin (0,0) a rectangle which has
base 12 and height 8.
In this final step pupils will add drawing the magic line
itself, connecting the Beetle’s and Dot’s positions.
❶ Pupils extend the behaviours of both sprites:
Whenever the Beetle draws its rectangle, the Dot
stamps its second (turquoise) costume at its position,
i.e. at the diagonally opposite corner of that rectangle
and switch the costume back to blue.
❷ Instead of drawing the whole rectangle in one colour,
pupils have the Beetle draw horizontal lines in a
different pen colour and pen size than the vertical
lines. Pupils modify the move horizontally and move
vertically blocks.
❸ Pupils build one more script for the Beetle: when
space key pressed, draw a “magic line” connecting the
Beetle’s and Dot’s positions.
❹ Pupils finalise the whole project so that it properly
works in the grid 10 backdrop.
LEARNING OBJECTIVES
Pupils continue in their own version of project 5-Grid World, or open project 5-Grid World
INT2, Save as a copy (online) or Save as (offline) and rename. The final version of this project at
the end of Activity 5.3.3 will be 5-Grid World FINAL.

55
INVESTIGATION 3
Ext. Activity 5.3.3
The key part here is this:
The Beetle first points towards the Dot, then slowly
starts moving by repeating move 5 steps, until it
touches the edge of the stage. Then it jumps back
home.
ADDITIONAL SUPPORT
❸
❷
❶

bridgE to knowledge of multiplication, regular polygons, angles, perimeter, ratio/proportion.
Envisage the outcome of different scripts.
Explain why a script would have a particular outcome and how to complete a script to generate
a specified outcome.
Print and distribute the unplugged pupil worksheet 5.3.4.
Ask pupils to work individually to check what they have learned during Module 5.
The answers to the worksheet are below:
1. For answer 20 the Beetle will say 40, for 1200 it will say 2400, for 45 it will say 90.
2. Second drawing.
3. The Beetle will ask only once, it will draw a regular octagon. If its perimeter is 160,
we must have answered the question by typing in 20.
4. Correct script is (b). Script (a) will draw gaps between the squares because it takes
the pen up each time it moves to the next square and script (c) does not increase
the side length at all so all the squares would be the same size.
5. First drawing Yes, if we answer 60. Each of three sides has the same length.
Second drawing No, as the sides have different lengths.
Third drawing Yes, if we answer 90.
Fourth drawing No, as it has four lines.
6. The outcome can be regular triangle, square, pentagon or hexagon.
7. 20; 40; 50; 80; 35; 60
8. It will draw 5 squares, the first one of 20, the last one of 60 side length. The value of
the variable at the end will be 70.
9. Script (a) produces the picture on the right.
Script (b) produces the picture in the middle.
Script (c) produces the picture on the left.
10. [Extension]
Proportionally bigger rectangles could be 25 and 100; 30 and 120; 40 and 16
Proportionally smaller rectangles could be 10 and 40; 5 and 20 and many others.
These are all rectangles with sides 1 : 4.
56
Unplugged: Module 5 Assessment
LEARNING OBJECTIVES

Read the tasks below and answer the questions.
Note - The pen tool is always down at the beginning of each task.
57
INVESTIGATION 3
Activity 5.3.4
NAME
WHAT TO DO
ASSESSMENT TASKS
❶ When we run this script, Beetle will ask for a
number. Note the number that the Beetle will say if
we give the following answers:
(note that * in Scratch means to multiply)
If we answer 20the Beetle will say
What will happen if we run this script and answer
the question “What pen size now?” by typing in 20?
Circle the correct drawing below.
(note the starting point is marked by the red arrow)
❷
❸ If we run this script:
a) How many times will the Beetle ask “what side length?”?
b) Describe what the Beetle will draw.
c) If the perimeter of the polygon that the Beetle draws is
160, what number was typed in?

58
ASSESSMENT TASKS CONTINUED
INVESTIGATION 3
Activity 5.3.4
Yes No
Why?
Yes No
Why?
Yes No
Why?
Yes No
Why?
For each of the following drawings decide whether it
can be an outcome of the script on the right or not.
Circle or tick Yes or No and explain why.
❹
❺
Circle the script that will produce the drawing below.
Explain why you picked that script:

59
INVESTIGATION 3
Activity 5.3.4
❼
❻ If we run this script for the Beetle what
kind of polygon is drawn?
Answer and explain your thinking:
For each of the following scripts of the Beetle
write down what side length the drawn square
will have:
side length of
square =
side length of
square =
side length of
square =
side length of
square =
side length of
square =
side length of
square =

60
INVESTIGATION 3
Activity 5.3.4
❿
❾ Match the script to the picture which it could produce.
❽
If we run this script…
How many squares will the Beetle draw?
What will be the side length of the first one?
What will be the side length of the last one?
What will be the value of the side length variable after the script is run?
[Extension] If we run script (a), the Beetle will draw the below rectangle. Think of two
more pairs of values of the height and base variables that would output a
mathematically similar bigger and smaller rectangle (i.e. fit on the magic line) and write
the numbers in the empty holes on the right (marked by ?).
proportionally bigger
proportionally smaller

61
Using the Grid World
This investigation uses the Grid World built in Investigation 3 to explore proportional
relationships and to solve problems which involve the mathematical ideas of ratio, proportion,
similarity and enlargement.
Pupils use the tool to represent the quantities within the problem as the two sides of a
rectangle, draw a magic line and then read off or calculate the quantities required in the
problem. Exploring pairs of values which fit on the magic line within the tool provides pupils
with an opportunity to develop their own connection within and between mathematically
similar rectangles. It is envisaged that the graphical representation be used along side other
representations within the classroom.
Activity 5.4.1 – Using the Grid World
Activity 5.4.2 – BridgEing And Solving Problems
Scratch starter project 5-Grid World
5-Grid World FINAL
Mathematics
Pupils recognise proportionality in contexts
when the relations between quantities are
in the same ratio (for example, similar
shapes and recipes).
Pupils consolidate their understanding of
ratio when comparing quantities, sizes and
scale drawings by solving a variety of
problems. They might use the notation a:b
to record their work.
Pupils solve problems involving unequal
quantities (for example, ‘for every egg you
need three spoonfuls of flour’)
► Pupils use the Grid World to explore and build
mathematically similar rectangles. They explain the
relationships between the lengths of sides in the same
ratio, within and between rectangles.
► Pupils solve various problems in contexts which require
proportionality. They work on and off the computer to
explain their solutions.

Explore how to build mathematically similar rectangles.
Explain mathematical relationships.
62
Using the Grid World
Pupils use their final 5-Grid World project, or the provided 5-Grid World FINAL project.
They use the grid 20 backdrop. A is the base of the rectangle and B is the height.
❶ Using the project, pupils construct a rectangle where A = 3 and B = 1.
❷ Pupils construct a rectangle where A = 6 and B = 2.
❸ Then they add the magic line.
❹ Pupils look for and construct two more rectangles which fit on the magic line.
❺ Does the rectangle where A = 15 and B = 5 fit on the line? Can you explain your answer?
❻ If A = 21, what is the value of B? Explain your answer.
❼ If B = 10, what is the value of A? Explain your answer.
❽ Pupils transfer their answers into the table on the printable worksheet 1 on the next page,
then they complete the calculations (solutions below).
LEARNING OBJECTIVES
A B A  B
3
6
21
36
10
1
2
3
3
3
3
3
3
3
12
9
30
12
4
3
7
WORKSHEET SOLUTIONS
❶

63
Transfer all your previous answers into the table, and complete the calculations.
INVESTIGATION 4
Activity 5.4.1
NAME
WHAT TO DO
WORKSHEET 1
A B A  B
3
6
21
36
10
1
2
❶

Explore how to build mathematically similar rectangles.
Explain mathematical relationships.
64
Pupils use their final 5-Grid World project, or the provided 5-Grid World FINAL project.
They use the grid 20 backdrop. A is the base of the rectangle and B is the height.
❶ Using the project, pupils construct a rectangle where A = 3 and B = 2.
❷ Pupils construct a rectangle where A = 6 and B = 4.
❸ Then they add the magic line.
❹ Pupils look for and construct two more rectangles which fit on the magic line.
❺ Does the rectangle where A = 12 and B = 6 fit on the line? Can you explain your answer?
❻ If A = 18, what is the value of B? Explain your answer.
❼ If B = 10, what is the value of A? Explain your answer.
❽ Pupils transfer their answers into the table on the printable worksheet 2 on the next page,
then they complete the calculations (solutions below).
LEARNING OBJECTIVES
A B A  B
3
6
18
30
10
2
4
1.5
1.5
1.5
1.5
1.5
1.5
1.5
12
9
15
20
8
6
12
WORKSHEET SOLUTIONS
❷
INVESTIGATION 4
Activity 5.4.1

65
Transfer all your previous answers into the table, and complete the calculations.
INVESTIGATION 4
Activity 5.4.1
NAME
WHAT TO DO
WORKSHEET 2
A B A  B
3
6
18
30
10
2
4
❷

66
INVESTIGATION 4
Activity 5.4.1
We can explore the constant relationships of mathematically similar rectangles in two ways:
• the relationship of the side lengths within a rectangle
• the relationship of one side length to a corresponding side length between two
rectangles
❶ Worksheet
The second worksheet looks very similar
to the first but is designed to encourage
the development of the relationship
between rectangles.
The relationship within each rectangle has
a constant ratio, base÷height = 3/2 = 1.5.
The base is 1.5 times bigger then the
height. The relationship is not as easy to
see or to calculate with as for whole
number ratios. However, performing
calculations between rectangles is easier.
Question 6: If A = 18, what is the value of
B? Explain your answer.
It can be difficult to calculate B since A is
1.5 times bigger than B. However, if we
know that a mathematically similar
rectangle is A=3 and B=2, then between
the rectangles there is a 6 times bigger
relationship, so if A=18 then B=2*6=12.
Question 7: If B = 10, what is the value of
A? Explain your answer.
Between the two rectangles there is a 5
times bigger relationship, so A=3*5=15.
❷
NOTE: The Grid World deals with whole numbers only, but we can observe that the ‘magic
line’ is an infinite line which passes through fractional and decimal parts of the grid.
Encourage pupils to calculate (they cannot check their work easily) with decimal values for
A and B in each of the above worksheets to develop fluency.
Encourage pupils to choose the method (within or between) which makes calculations
easier. Ask them to select which method they use when solving problems.
Worksheet
This worksheet is designed to encourage the
development of the relationship within a
rectangle, i.e. the multiplicative relationship
between base (A) and the height (B).
The ratio can be calculated, base÷height = 3.
It is important to recognize that the inverse
ratio height÷base is also a constant 1/3.
We can write:
• base = height x 3, or
• A = B x 3, or
• B = A/3 B = 1/3 of A
Question 6 is small enough to fit on the grid,
however pupils should be encouraged to
calculate first and then test their solution in
the Grid World. Question 7 cannot be tested
on grid 20 but can be checked by switching
to grid 10 .

67
BridgEing And Solving Problems
Activity 5.4.2 consists of two worksheets. Print and distribute them or work as a class.
Pupils use their final 5-Grid World project, or the provided 5-Grid World FINAL project to solve
the problems. Encourage pupils to sketch rectangles.
Discuss solutions and approaches as a class. Additional notes in solutions.
1. The base of the first rectangle is twice a big as the height. Therefore for a mathematically
similar rectangle which has a height of 3, the base must be 3 x 2 = 6
[This is an example of a within relationship]
2. The relationship of the matches to bottle tops can be thought of as comparing the sides of
a rectangle. Draw a diagram of a 2 x 3 rectangle to help pupils.
It is easier to use a between relationship.
12 matchsticks is 6 times bigger than 2 matchsticks, so we need 3 x 6 = 18 bottle tops.
3. One approach to solve this problem is to combine two similar rectangles. We’ve seen in
the growing rectangle sequence that if you change the height and base in the same
proportion you make a similar rectangle.
Mr Short is 4 buttons and 6 paperclips, we want 6 buttons.
We could find the relationship within (2/3) or between (3/2) but both of these are
difficult to calculate with.
An easier approach is to half both, so 2 buttons and 3 paperclips.
We can now add together since we are adding in the same ratio (proportion)
So 6 buttons tall is the same as 6 + 3 = 9 paperclips tall. Mr Tall is 9 paperclips tall.
LEARNING OBJECTIVES
WORKSHEET SOLUTIONS
bridgE to problems which involve the mathematical ideas of ratio, proportion and similarity.

68
BridgEing And Solving Problems
4. This problem is adapted from a task within the http://iccams-maths.org/ project.
a) Although this a recipe problem, we can still use Grid World to help us compare two
quantities and to find quantities which have the same relationship by using the “magic
line”.
The two sides of the rectangle can represent any two quantities, in this problem the base
represents the quantity of tabasco sauce and the height represents the number of people.
Pupils draw:
b) Pupils draw a rectangle which has a height of 6, the base will be 18. This will fit on the
magic line.
c) There are many solutions, the amount of Tabasco is always 3 x the number of people.
The grid default grid word will show (3, 1), (6,2), (12, 4), (15, 5).
d) The amount of Tabasco = 3 x the number of people.
e) For 30 people, you will need 3 x 30 = 90ml of Tabasco.
f) 57/3 = 19
19 people could be served.
WORKSHEET SOLUTION CONTINUED

69
INVESTIGATION 4
Activity 5.4.2
NAME
WHAT TO DO
Use the Grid World to solve the problems, draw diagrams to explain your working.
WORKSHEET
The two rectangles are proportional (mathematically similar) to one another.
Find length x and give a reason for your answer.
❶
❷ Two matchsticks have the same length as three bottle tops.
How many bottle tops will have the same length as 12 matchsticks?
Answer and explain your thinking
❸ Here is a picture of Mr. Short. Mr Short is 4 buttons or 6 paperclips in height. Mr Tall
measures 6 buttons, how high is Mr Tall in paperclips?

70
INVESTIGATION 4
Activity 5.4.2
NAME
WHAT TO DO
Use the Grid World to solve the recipe problem.
WORKSHEET
Adam is making a spicy soup for 3 people. He uses 9ml of tabasco sauce.
a) Create a rectangle in the Grid World which has a base of 9 and a height of 3,
draw the magic line.
b) Davina is making the same soup for 6 people. How much tabasco sauce should
she use?
[Clue: Draw a rectangle which fits on the magic line and has a height of 6.]
c) Find another solution that works (i.e. fits on the magic line).
d) What is the relationship between pairs of numbers?
e) If Matt is making the same soup for a large party of 30, how much tabasco
would he need?
f) Tabasco is sold in 57ml bottles. How many people could be served the same
spiciness of soup using one bottle?
❹

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