This week I will mostly be playing with Probots and the Educational freeware environment MSWLOGO I had installed in school a while back. LOGO is an environment I haven't explored in as much depth as I should have, or so I am beginning to discover, particularly with the increased focus within the New Primary Framework for Mathematics on the using and applying strands.
As I have begun to play with these tools this week, exploring beyond the simple repeat and build procedures I initially work through with students, I am beginning to discover just how superficial my personal understanding and capability with the tool is, and just how big a mistake it was on my part not to persist with the environment in my early teaching career. These short sharp sessions are beginning to challenge me to think more deeply about how I could use tools like LOGO more regularly to challenge and support reasoning and thinking with students during the numeracy hour, a place where I have long believed this aspect of Control and modelling within the primary ICT curriculum should be embedded.
Experience tells me there is a tendency for LOGO to be used and taught in the Primary School these days when the QCA Units for ICT that utilise it make that requirement. This leads generally to control sessions being developed through the delivery of standalone units such as this in Y4. What concerns me about this is not children learning to build procedures that make pretty flower designs, but that from a pedagogical perspective there is much more to LOGO as a learning tool than this. By linking the tool to help develop units of work involving application of knowledge and understanding of shape, space and measure. As a "constructivist" learning tool, LOGO affords opportunities to design learning contexts through which to apply, link and develop strategies for problem solving and investigational work while using the environment to model and explore generalisations and hypotheses.
Thinking about Rectangles
While playing today I have been thinking about rectangles and how, when shown an oblong many of my students are able to identify and describe its basic properties, eg. it is a rectangle, it has four sides, four angles or corners, and each angle is a right angle. When we compare the shape with a square that shares these properties, and the students are asked how they differ? discussions usually begin with how one shape is longer than the other. Further questioning and discussion will lead to us identifying how opposite sides of an oblong are the same length, and how the square is a special (regular) rectangle, but making specific reference to the nature of the opposing sides of an oblong is something which is quickly forgotten.
Coding LOGO to draw an oblong requires us to use and apply this property. Even though the shape has 4 sides, a repeat 4 procedure, would not be an efficient way to input the parameters. Writing even a simple repeat procedure, requires us to think slightly differently about the way we should input our commands in order to draw the oblong. So how might I use LOGO to explore and consolidate understanding about the properties of rectangles as members of the quadrilateral family.
Thinking Through an Investigation
In previous sessions, adapting the QCA Unit linked to above while working with students I have tended to stick to developing regular polygons of a standard side length, returning to making Irregular Rectangles or oblongs later, perhaps if or when students have asked how to do this. This leads to a naturalish discussion about the properties of rectangles.
As said above squares can be classified as Regular Rectangles, as such they have 4 equal sides, 4 equal and identical angles, and the same number of lines of symmetry as they do sides and angles. Each angle is a right angle measuring 90 degrees.
A square can be made with a repeat procedure
repeat 4 [fd X rt 90]
Or using the polygon procedure I published previously, by inputting polygon :side :length.
This would not work for an oblong since adjacent sides are different lengths, while opposite sides are equal.
Building a Procedure to Draw Rectangles
Lets say we wanted the turtle to travel around the perimeter of an oblong 300 units long, we could do this by inputting commands that would draw 2 sides of 100 units and 2 sides of 50 units, using a set of commands that look something like this
fd 100 rt 90 fd 50 rt 90 fd 100 rt 90 fd 50 rt 90
To turn this into a repeat procedure, we could rewrite it something like this:
repeat 2 [fd 100 rt 90 fd 50 rt 90]
So a build command would look something like this...
repeat 2 [fd 100 rt 90 fd 50 rt 90]
Saving this procedure, typing in rectangle and hitting the enter key would draw a rectangle where 2 sides were 100 units long and 2 sides were 50 units long.
Building a Procedure to Create Variable Rectangles
The last procedure would be fine if we always wanted to make our rectangle the same shape and size, or we were happy to edit our procedure every time we wanted to change the size of our rectangle. But if we wanted to use our program to help us investigate for example the perimeters of rectangular fields that a farmer could fence with 300 metres of wire? Or the different rectangles we could draw with a particular perimeter it would be useful to have a program where we were able to substitute parameters. (Thanks Andy am having lots of fun with this!) In this case we might use a procedure something like this:
to rectangle :sidea :sideb
repeat 2 [fd :sidea rt 90 fd :sideb rt 90]
Investigating Rectangles Maybe!
Now I could begin either inputting side lengths to the procedure to randomly draw different rectangles, or I could begin applying my knowledge and experience about the properties of rectangles, using other areas of mathematical experience such as calcualation, addition and subtraction, halving and doubling etc to help me explore/investigate the substitution values that total a given perimeter, using my procedure to model and test my work by inputting my values and testing on screen. Eg with my length of 300 metres or Turtle units
rectangle 40 110 and hitting the enter key would draw on oblong with this perimeter
The procedure models the familiar formula for calculating the perimeter of a rectangle
(2x40) + (2x110) = 300
What about rectangle 45 105?
A suitable problem might be for children to investigate some of the different ways of fencing a rectangular field with 300 metres of wire (using 300 turtle units to represent this).
I don't think I would show the children the procedure and how it works. Instead it might be fun to begin with trial and error processes through the introduction and investigation of a systematic pattern, similar to the example below
40+40+110+110= 300 (rectangle 40 110)
45+45+105+105=300 (rectangle 45 105)
50+50+100+100=300 (rectangle 50 100)
What might the next sequence or input be? What can we see happening in the pattern? What would happen to the pattern if we changed 40 to 41 or 42 and so on? The students could use whiteboards to record their patterns, inputs and informal jottings if necessary, with the focus of the task being around paired discussion, mediated by the onscreen feedback and success of their input in relation to the expected output of a rectangle.
Moving on perhaps we could ask the children to use their experiences to visualise, map out and draw what they think the turtle has to do in creating each of the rectangles they have drawn, focusing their attention to the visible properties of each shape and how these relate to the numerical values they have inputted. Notice the line (perimeter) made by the turtle travels around the shape, so each step carried out must have been in sequence. Even though still focussed on the properties of oblongs and the relative location of their sides, we also have the contingency within this activity to begin to introduce the idea that 2 of the values in each of the patterns are the same, and that as the turtle draws them they output to become opposite sides of the oblong and that adjacent sides are of different lengths. So perhaps at this point we could look at or introduce a simple repeat procedure for drawing a rectangle and explore how each value might be substituted into the program, talking through the effect, drawing and visualising it before returning trying it on screen, perhaps with trace enabled, and finally reviewing our square and oblong, discussing how or if what we have learned during our investigation has changed our view of the two shapes. This might also support introduction of the formula for calculating perimeter of squares and other rectangles, by comparing and deriving the standards from the LOGO procedures.
eg repeat 4 [fd x rt 90]
could be reduced to the perimeter of a square is equal to 4 times the length of 1 side
repeat 2 [fd x rt 90 fd y rt 90]
could be reduced to the perimeter of a rectangle is equal to 2 times the length add 2 times its width.
Using the rectangle procedure again the task could be extended to enable students to test and investigate the shapes of fields that can be fenced using different lengths of wire by the farmer. What length would each side be if the farmer wanted to make a square field using the length of wire he has?
In class and at table tops, this activity could be further extended to explore the relationship between area and perimeter in each shape, using squared paper to explore the different shaped fields the farmer can make, and the area of each one.
This may seem an ambitious proposal, but I would be fascinated to see just what a KS2 class would make of the activity. I think most of us would be surprised to see just what a Y4 class can learn about the properties of polygons and rotation when using the idea that 360 degrees is the same as a full turn, and substituting factors of 360 within repeat procedures to make "flowers" from fixed shape procedures. The potential mathematical outcome of the Y4 unit linked to above, is way more complex than on first viewing the unit expects, yet if students are to appreciate and understand the outcomes of the tasks, beyond the making of flowers and context is to be given to what is being learned we need to engage the children with the mathematical ideas inherent within it. Perhaps this unit can be taken to pieces, and bits and pieces used or built into wider mathematical work. How might changing our view of this unit enable exploration and application of visual models to support reasoning and understanding about shape?