UCL ScratchMaths Curriculum Exploring Mathematical Relationships

UCL ScratchMaths Curriculum Exploring Mathematical Relationships
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 ( [CC BY 2.0 ( via Wikimedia Commons  Top right: Sean Macintosh [CC BY-SA 4.0 ( via Wikimedia Commons  Bottom left: Francisco Anzola (Doha skyline in the morning) [CC BY 2.0 ( via Wikimedia Commons  Bottom right: Wilson Hui from Calgary, Canada - Shanghai Skyline from the Bund, CC BY 2.0,

Module 5: Exploring Mathematical Relationships
Investigation 1 Polygon Fireworks Night Skyline Investigation 2 Mathematically Similar Rectangles Investigation 3 Grid World: For Exploring Similarity Investigation 4 Exploring Proportionality 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. Key Vocabulary and Concepts Covered by Module 5 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

Polygon Fireworks, Night Skyline
Map Of Module 5 Activity 1 Activity 2 Activity 3 Activity 4 Investigation 1 Polygon Fireworks, Night Skyline Ask and Answer Starter project: 5-Polygon Firework Unplugged: Polygon Predictions Naming Values Continue with: or start with: 5-Polygon Firework INT1 The Sky at Night 5-Polygon Firework INT2 Investigation 2 Mathematically Similar Rectangles Sequences of Squares Starter project: 5-Altering Polygons Altering Rectangles Continue with: Exploring Mathematical Similarity Unplugged: Rectangle Jungle Activity 1 Investigation 3 Grid World: For Exploring Similarity Enter the Grid World Starter project: 5-Grid World Connecting Corners Continue with: or start with: 5-Grid World INT1 Meet the Magic Line 5-Grid World INT2 Unplugged: Module 5 Assessment Investigation 4 Exploring Proportionality Using the Grid World Continue with: 5-Grid World or start with: 5-Grid World FINAL Activity 4 BridgEing and Solving Problems Continue with: 5-Grid World or start with: 5-Grid World FINAL 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.

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.

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 – Ask and Answer Activity – Unplugged: Polygon Predictions Activity – Naming Values Activity – 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 Curriculum Objectives Link with ScratchMaths 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.

Module 5 ● Investigation 1 ● Activity 5.1.1 Ask and Answer
Learning Objectives 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. Activity Instructions 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 will be 5-Polygon Firework INT1. 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 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. 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°]

Investigation 1 …continued Activity 5.1.1
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.

❶ ❸ ❽ Investigation 1 Connections To Y5 ScratchMaths Activity 5.1.1
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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: ❸ In Module 2, Activity 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. ❽ In Module 2, Activity 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.

❶ ❷ Investigation 1 Additional Support Activity 5.1.1
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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). ❷

Investigation 1 Activity 5.1.1 Additional Support Continued ❸ ❹ ❺ ❻ ❼

❽ Investigation 1 Additional Support Continued 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.

Unplugged: Polygon Predictions
Module 5 ● Investigation 1 ● Activity 5.1.2 Unplugged: Polygon Predictions Learning Objectives 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. Activity Instructions Print and distribute the pupil worksheet 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. Solution ❶ 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. ❷ 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. ❸ 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. ❹ 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. ❺ 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.

Investigation 1 ❶ ❷ ❸ ❹ ❺ Name What To Do Ask Answer Scripts
Activity 5.1.2 Name What To Do 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. Ask Answer Scripts ❶ ❷ ❸ ❹ ❺

❶ ❷ ❸ ❹ Investigation 1 Extension Activity Instructions
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. ❶ ❷ ❸ ❹ Additional Support ❶ 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. ❷ ❸ ❹

Module 5 ● Investigation 1 ● Activity 5.1.3 Naming Values
Learning Objectives 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. Activity Instructions Mathematics Connections 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 will be 5-Polygon Firework INT2. 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. You may prefer to do most of this activity (up to point 4, including) using the empty plain backdrop. Note that the actual setting of a variable happens only after you run the block – by clicking or running a script containing that block.

❶ ❷ ❸ Investigation 1 Additional Support Activity 5.1.3
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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). ❷ 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

❹ ❺ Investigation 1 Additional Support Continued 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. ❺ 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.

Choosing names for variables
Investigation 1 Activity 5.1.3 Additional Support Continued 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… Extension Ideas Explore the following Surprising polygons: 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°.

Module 5 ● Investigation 1 ● Activity 5.1.4 The Sky at Night
Learning Objectives 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. Activity Instructions Mathematics Connections 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 will be 5-Polygon Firework FINAL. 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?

❶ Investigation 1 Connections To Y5 ScratchMaths Activity 5.1.4
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ In Module 2, Activity pupils drew a square and a equilateral triangle. In Activity they were encouraged to give a name to their square script, making their own square block. In Activity 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. 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 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.

❶ ❷ ❸ ❹ Investigation 1 Additional Support Activity 5.1.4
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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 … ❷ ❸ ❹

❺ Investigation 1 Additional Support Continued 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

Module 5: Investigation 2 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 –Sequence of Squares Activity – Altering Rectangles Activity – Exploring Mathematical Similarity [Extension] Activity – Unplugged: Rectangle Jumble Scratch starter project 5-Altering Polygons Links to Primary National Curriculum Curriculum Objectives Link with ScratchMaths 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.

Module 5 ● Investigation 2 ● Activity 5.2.1
Sequence of Squares Learning Objectives 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. Activity Instructions Mathematics Connections Pupils open project 5-Altering Polygons, Save as a copy (online) or Save as (offline) and rename. 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. 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.

❸ Investigation 2 Connections To Y5 ScratchMaths Activity 5.2.1
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❸ Within the 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. 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). For example in Module 3, Activity we used change y by … block to make Tera jump high then slowly float back:

❶ ❷ ❸ ❺ ❹ Investigation 2 Additional Support Activity 5.2.1
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ ❷ ❸ Click script (a) again and again. Observe the value of side length. Then click script (b) several times and observe. ❹ ❺ Below see alternative solution of the same task. Discuss it with the pupils.

❻ Investigation 2 Additional Support Continued 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? ❻ Two alternative solutions of 6 (a). 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.

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

Investigation 2 Extension Ideas 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.

Altering Rectangles 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. Activity Instructions Mathematics Connections Pupils continue in their own version of project 5-Altering Polygons, or start again with the initial starter project 5-Altering Polygons. 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. 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.]

❶ ❷ ❸ Investigation 2 Additional Support Activity 5.2.2
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ Encourage pupils to spot the regularity in drawing the rectangle and ensure the script reflects that regularity. ❷ 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. ❸

Investigation 2 Activity 5.2.2 Additional Support Continued ❹ ❺

A slide from the pupil presentation for activity 5.2.2.
Investigation 2 Activity 5.2.2 Additional Support Continued 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. A slide from the pupil presentation for activity 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.

Investigation 2 Extension Ideas Activity 5.2.2
Challenge pupils to create interesting outcomes using the rectangle block inside a repeat.

Exploring Mathematical Similarity 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. Activity Instructions Mathematics Connections Pupils continue in their own version of project 5-Altering Polygons, or start again with the initial starter project 5-Altering Polygons. In Activity 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). 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].

❶ ❷ Investigation 2 Additional Support Activity 5.2.3
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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. ❷ 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. 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. 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.

❸ Investigation 2 Additional Support Continued Activity 5.2.3
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. 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. h b 90 30 100 20 95 45 120 60 25 15

Investigation 2 Name What To Do Sequence Of Rectangles A
Activity 5.2.3 Name What To Do 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. Sequence Of Rectangles A Sequence Of Rectangles B 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Sequence Of Rectangles C Sequence Of Rectangles D 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Investigation 2 Worksheet Solutions Sequence Of Rectangles A
Activity 5.2.3 Worksheet Solutions 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. Sequence Of Rectangles A Sequence Of Rectangles B 1 2 1.67 3 1.50 4 1.40 5 1.33 6 1.29 7 1.25 8 1.22 1 2 2.50 3 4 3.50 5 6 4.50 7 8 5.50 Sequence Of Rectangles C Sequence Of Rectangles D 1 2 1.33 3 4 0.80 5 0.67 6 0.57 7 0.50 8 0.44 1 2 3 4 5 6 7 8

Investigation 2 Name What To Do Sequence Of Rectangles E
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. Sequence Of Rectangles E Sequence Of Rectangles F 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Sequence Of Rectangles G Sequence Of Rectangles H 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

[Extension] Unplugged: Rectangle Jumble Learning Objectives Envisage which rectangles are proportional to one another. Explain why two rectangles are proportional. bridgE to knowledge of ratio and proportion. Activity Instructions Activity 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. Worksheets Solutions ❶ All rectangles proportional to the swimming pool 60 metres by 15 metres are highlighted. There are three groups of proportional rectangles ❷ ❸ For the third worksheet the solution is 1 – 4, 2 – 5, 3 – 1, 4 – 2, 5 – 3.c

❶ Investigation 2 Name What To Do Swimming Pool Plans
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

❷ Investigation 2 Name What To Do Rectangle Jumbles
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. Rectangle Jumbles Sort the rectangles into three groups (2 to 1, 3 to 1 and 4 to 1) the jumble below.

❸ ❶ ❷ ❸ ❹ ❺ Investigation 2 Name What To Do
Ext. Activity 5.2.4 Name What To Do 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. Match Scripts Between Columns ❶ ❷ ❸ ❹ ❺

Module 5: Investigation 3 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 – Enter The Grid World [Extension] Activity – Connecting Corners [Extension] Activity – Meet the Magic Line [Extension] Activity – Unplugged: Module 5 Assessment Scratch starter project 5-Grid World 5-Grid World INT1 5-Grid World INT2 Links to Primary National Curriculum Curriculum Objectives Link with ScratchMaths 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.

Module 5 ● Investigation 3 ● Activity 5.3.1 Enter The Grid World
Learning Objectives Envisage the size of a grid tile using coordinates. Explore how to move horizontally and vertically by a specified number of “tiles”. Activity Instructions Mathematics Connections 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 will be 5-Grid World INT1. 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… 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. 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 ❷ ❸ ❹ Investigation 3 Additional Support Activity 5.3.1
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ The actual coordinates of the mouse cursor within the stage are displayed under the lower right corner of the stage. ❷ ❸ The Beetle may move by 1 tile either in one go (one move) or by repeating smaller step several times. ❹ 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.

❺ ❻ ❼ Investigation 3 Additional Support Continued Extension
Activity 5.3.1 Additional Support Continued ❺ 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. ❻ 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. Extension ❼ Here are different definitions of the move 1 tile block. Envisage which of them will make the Beetle move very quick, very slow, slow…

[Extension] Connecting Corners
Module 5 ● Investigation 3 ● Activity 5.3.2 [Extension] Connecting Corners Learning Objectives 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. Activity Instructions Mathematics Connections 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 will be 5-Grid World INT2. 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. 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.

❶ ❷ ❸ ❹ ❺ Investigation 3 Additional Support Ext. Activity 5.3.2
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ 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. ❷ 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. ❸ 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. ❹ Discuss with your class which arrows affect variable A and how and which arrows affect variable B. ❺

❻ ❼ Investigation 3 Additional Support Continued
Ext. Activity 5.3.2 Additional Support Continued ❻ 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 ❼ 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.

❽ Investigation 3 Additional Extension Support Continued
Ext. Activity 5.3.2 Additional Extension Support Continued ❽ 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.

[Extension] Meet the Magic Line
Module 5 ● Investigation 3 ● Activity 5.3.3 [Extension] Meet the Magic Line Learning Objectives Explore how to draw the “magic line”. Activity Instructions Mathematics Connections 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 will be 5-Grid World FINAL. 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. 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.

❶ ❷ ❸ Investigation 3 Additional Support Ext. Activity 5.3.3
Please note the blue numbers on the left link to the numbered steps in the activity instructions ❶ ❷ ❸ 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.

Module 5 ● Investigation 3 ● Activity 5. 3
Module 5 ● Investigation 3 ● Activity Unplugged: Module 5 Assessment Learning Objectives 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. Activity Instructions Print and distribute the unplugged pupil worksheet Ask pupils to work individually to check what they have learned during Module 5. The answers to the worksheet are below: For answer 20 the Beetle will say 40, for 1200 it will say 2400, for 45 it will say 90. Second drawing. 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. 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. 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. The outcome can be regular triangle, square, pentagon or hexagon. 20; 40; 50; 80; 35; 60 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. Script (a) produces the picture on the right. Script (b) produces the picture in the middle. Script (c) produces the picture on the left. [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.

❶ ❷ ❸ Investigation 3 Name What To Do Assessment Tasks Activity 5.3.4
Read the tasks below and answer the questions. Note - The pen tool is always down at the beginning of each task. 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 20 the Beetle will say If we answer 1200 the Beetle will say If we answer 45 the Beetle will say ❷ 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) If we run this script: How many times will the Beetle ask “what side length?”? Describe what the Beetle will draw. If the perimeter of the polygon that the Beetle draws is 160, what number was typed in? ❸

❹ ❺ Investigation 3 Yes No Yes No Yes No Yes No
Activity 5.3.4 Assessment Tasks Continued ❹ Circle the script that will produce the drawing below. Explain why you picked that script: ❺ 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. Yes No Why? Yes No Why? Yes No Why? Yes No Why?

❻ ❼ Investigation 3 Assessment Tasks Continued Activity 5.3.4
If we run this script for the Beetle what kind of polygon is drawn? Answer and explain your thinking: ❼ 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 =

❽ ❾ ❿ Investigation 3 Assessment Tasks Continued Activity 5.3.4
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? ❾ Match the script to the picture which it could produce. ❿ [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

Module 5: Investigation 4 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 – Using the Grid World Activity – BridgEing And Solving Problems Scratch starter project 5-Grid World 5-Grid World FINAL Links to Primary National Curriculum Curriculum Objectives Link with ScratchMaths 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.

Module 5 ● Investigation 4 ● Activity 5.4.1 Using the Grid World
Learning Objectives Explore how to build mathematically similar rectangles. Explain mathematical relationships. Activity Instructions ❶ 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). Worksheet Solutions A B A  B 3 1 3 6 2 3 12 4 3 9 3 3 21 7 3 30 10 3 36 12 3

❶ A B A  B 3 1 6 2 21 10 36 Investigation 4 Name What To Do
Activity 5.4.1 Name What To Do Transfer all your previous answers into the table, and complete the calculations. Worksheet 1 A B A  B 3 1 6 2 21 10 36

Investigation 4 Activity 5.4.1 Learning Objectives Explore how to build mathematically similar rectangles. Explain mathematical relationships. Activity Instructions ❷ 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). Worksheet Solutions A B A  B 3 2 1.5 6 4 1.5 12 8 1.5 9 6 1.5 18 12 1.5 15 10 1.5 30 20 1.5

❷ A B A  B 3 2 6 4 18 10 30 Investigation 4 Name What To Do
Activity 5.4.1 Name What To Do Transfer all your previous answers into the table, and complete the calculations. Worksheet 2 A B A  B 3 2 6 4 18 10 30

❶ ❷ Investigation 4 Mathematics Connections 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 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 . 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 = 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.

BridgEing And Solving Problems Learning Objectives bridgE to problems which involve the mathematical ideas of ratio, proportion and similarity. Activity Instructions Activity 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. Worksheet Solutions 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] 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. 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 = 9 paperclips tall. Mr Tall is 9 paperclips tall.

BridgEing And Solving Problems Worksheet Solution Continued This problem is adapted from a task within the 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 = people could be served.

❶ ❷ ❸ Investigation 4 Name What To Do Worksheet Activity 5.4.2
Use the Grid World to solve the problems, draw diagrams to explain your working. Worksheet ❶ The two rectangles are proportional (mathematically similar) to one another. Find length x and give a reason for your answer. 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? Answer and explain your thinking

❹ Investigation 4 Name What To Do Worksheet Activity 5.4.2
Use the Grid World to solve the recipe problem. Worksheet ❹ 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? Answer and explain your thinking

UCL ScratchMaths Curriculum Exploring Mathematical Relationships

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