Sunday, 22 April 2018

ALG 11

THIS WEEK: We explore the phenomenon of Letter as Object, something we met briefly in ALG 9C, ie we look at contexts where a letter represents a pure number (of objects) or a quantity (the price or length or mass or some other numerical quality of an object), but where there might be a temptation to use the letter as a shorthand for the object itself (as in 'a stands for apple, b stands for banana' - the classic fruit salad algebra). We will see that sometimes one can slide harmlessly between letter as object and letter as quantity, but that sometimes it leads to a completely fake algebra.
MONDAY: In this pair of tasks m stands for a numerical quality (mass) in the first task and for a pure number in the second. Though the answer is 5m in both cases, we have found that the second kind of task is substantially more demanding than the first - does that hold for your students?
In the first task, it is easy to interpret m as simply standing for mints rather than the mass of a mint, but still come up with the correct expression, 5m, albeit misinterpreted as 5 mints. In the second task students have to cope with the idea that m is most definitely a number, but one whose value we don't know, so that it is not possible to arrive at a specific numerical answer.
TUESDAY: Careful! Needles and pins. Because of all my pride, the tears I gotta hide... This turns out to be quite a Searching task, though let's hope it doesn't quite lead to tears!
The task involves an expression in which the letters stand for numbers but where the temptation to treat them as objects is very strong (for us as well as our students) and where this leads to a complete misinterpretation of the expression.  
Why can't I stop
And tell myself I'm wrong
I'm wrong, so wrong
Why can't I stand up
And tell myself I'm strong

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The extract below is from my article Object lessons in algebra? that appeared in Mathematics Teaching 98 in 1982. Still worth reading!
WEDNESDAY: In this task, the misinterpretation that results from treating the letters as objects is not as jarring... The given expression stands for the cost (in number of pence) of 5 bananas and 2 coconuts, so it is wrong to simply translate it as 5 bananas and 2 coconuts. Nevertheless this translation does make some kind of sense - the story is about 5 bananas and 2 coconuts, whereas the corresponding interpretation of the expression 2p + 5n in ALG 11B, as 2 pins and 5 needles, does not fit that story at all.
It is interesting to consider the alternative version of task 11C shown below.

Here students are quite likely to come up with the right expression, though not necessarily for the right reason - we can't be sure whether students who write 5b + 2c fully realise that this represents a number of pence and that it is not simply telling us about the number of bananas and coconuts bought. So this alternative version is not as useful as the original for revealing students' thinking. The task below, which appeared recently on Twitter, is even less effective as a diagnostic tool: here the correct option (B, I assume) can be chosen by simply treating it as an abbreviation of the verbal statements, ie by reading 5t + 2c = 3.70 as 5 teas and 2 coffees cost 3.70 (pounds), rather than appreciating that 5t + 2c actually represents the number of £s spent on the drinks.
THURSDAY: We present the first of two tasks involving sets of coloured rods, which we ask students to symbolise in different ways.

x

Sunday, 15 April 2018

ALG 10

This week we describe various situations using symbolic algebra. We then 'play' with the algebra by choosing values that take us beyond the immediate situation: can we relate the algebra back to the situation? Does the situation still make sense?
The situations we've chosen are fairly straightforward, but the game we're playing is mathematically quite sophisticated. It's similar to starting with a familiar statement like 8 + x = 10, which works in natural numbers, and asking what happens in a case like 8 + x = 5: this can be made to work if we stretch our ideas about number, ie if invent new ones - the integers.
Note: From an RME perspective, we could say that we are engaged in horizontal mathematisation (expressing a 'real' situation mathematically) and then in vertical mathematisation (developing the maths).
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MONDAY: Here we play with an area formula, for a shape that can vary in size. We start with a straightforward application of the formula but then consider a case which only works if we allow (or invent) edges with negative lengths.
Some students might feel that negative lengths are simply not allowed. That is a perfectly defensible position, but it would restrict the mathematics that we are able to do. A simple response is to say we are going to enter (or invent) a new (mathematical) world where negative lengths are allowed. So there!
This is what the shapes in parts i and ii look like (if you allow negative lengths in part ii):
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TUESDAY: We look at another familiar area scenario, namely area of a trapezium (and, to keep it simple, a trapezium that is right-angled).
Here we need to accept the idea of a negative length again, but also the idea of a negative area (which we could sidestep or leave implicit in ALG 10A).
One way to find the area geometrically is to divide the trapezium into two triangles. For part i we can, for example, divide the trapezium into two triangles of area 30 and 15 square units (top-left diagram, below).
The top-right diagram shows what happens to the trapezium as point P moves until it is 5 units to the left of the formerly top-left vertex. The trapezium 'twists' over itself to form two triangles whose areas, we can argue, are 20 and -5 square units.
The two diagrams at the bottom of the slide, below, show an alternative interpretation for part ii. The yellow triangle corresponds to the 30 square units triangle in the top-left diagram. The green triangle corresponds to the 15 square units triangle in the same diagram, except its base has changed from 5 units to -5 units. If we 'cancel' the region where the yellow and green triangles overlap, we are left with the regions with area 20 and -5 square untis shown in the top-right diagram.
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WEDNESDAY: Things get really interesting.... What's a 2-and-a-half sided regular polygon?! 
I came across this beautiful idea, of replacing the whole number of sides, n, with a fraction, in one of David Fielker's articles in Mathematics Teaching, many years ago. For me, the idea is almost on a par with inventing negative or fractional indices. A simple but brilliant mathematical act!
You may recognise the form of the instructions if you are familiar with LOGO and Turtle Geometry. If you don't have a turtle to hand I hope you will have enacted the instructions yourself and traced the resulting paths on paper or in your head! This is what they turn out to look like:
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THURSDAY: We again make the shift from whole numbers to fractions, this time on a familiar number grid.
 It can be useful to consider what happens to the sum, S, when the T-shape moves across the grid (eg 1 square to the right, or 1 square up). We can think about this spatially (What happens to each of the numbers in the T-shape?) or algebraically (What happens to S when n in the expression 6n+120 is increased by 1, say, or by 10?). In the case of part i, S has increased by 240–210 = 30, which can be achieved by moving the T-shape 5 squares to the right... [Are there other ways?]
Here are positions for the T-shape for parts i and ii.
Note: if we accept the principle behind the part ii answer, of allowing fractions of a square, we can find infinitely many positions for the T-shape, for parts i and ii, by moving the shape vertically (maybe just a tiny bit) as well as horizontally. What we're doing here, in effect, is to change the discrete 2-D grid into a continuous Cartesian plane.
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FRIDAY: Here we consider square grids made of matchsticks - or parts of matchsticks.
I'm particularly fond of this pattern - it's one whose structure is fairly easy to discern generically, even though the relation between the dimension of the square and the number of matchsticks is quadratic rather than linear.
Note: The slide below shows some interesting attempts to structure the grid by three Year 7 students (from a 'low attaining' set: set 3 of 4). I've written this up in chapter 3 of the Proof Materials Project report, Looking for Structure.
Here's a solution to ALG 10E (below). The last part is, of course, the most interesting. Using an expression for the number of matchsticks for an n by n grid, we get 31.5 sticks for a 3.5 by 3.5 grid. We've constructed a drawing for the grid that fits that total by allowing fractional matchsticks - though it's up to you whether you are willing to accept this! The drawing consists of 24 whole sticks, 8 sticks split in half 'cross ways', another 6 sticks split in half length ways, and two quarter sticks (resulting from being split in half cross ways and length ways). This makes 24 + 4 + 3 + 0.5 sticks = 31.5 sticks. 
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NEXT WEEK: We revisit the phenomenon of Letter as Object, ie we look at contexts where a letter represents a pure number (of objects) or a quantity (the price or length or mass or some other numerical quality of an object), but where there might be a temptation to use the letter as a shorthand for the object itself (as in 'a stands for apple, b stands for banana' - the classic fruit salad algebra). We will see that sometimes one can slide harmlessly between letter as object and letter as quantity, but that sometimes it leads to a completely fake algebra.

Sunday, 8 April 2018

ALG 9

As it's not quite the summer term yet (some schools go back on Monday, some on Monday week), the theme this week is not quite algebra. We are going to look at some tasks from well-known (or well-publicised) textbooks that involve faux, fake, phony, manqué, mock, pretend, pseudo, quasi, cod algebra.
MONDAY: a peach from Singapore (or several peaches, if you can afford them).
What is going on here? Does the story make sense?
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TUESDAY: We're still in Singapore, with a task where x is not so much an unknown as an utter mystery. What on earth might it stand for?
Mei Heng has unusual powers. The longer she works, the faster she works. However, if she works for less that 3 hour 40 minutes she seems to destroy rather than make T-shirts. Perhaps the Singapore government has created a mechanism that discourages part-time employment...
What of her pay, I wonder. Is it per hour or per T-shirt?
NOTE: I've written about this and a few other tasks from the same textbook in the journal Mathematics in School (November 2013, Volume 42, Issue 5, p25).
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WEDNESDAY: The context here involves 'grids' made of rods joined by a variety of links. We are shown two specific grids and for each we are presented with a 'rule' stating how many rods and various links it consists of.
The 'rules' look 'algebraic' in that they contain letters. However, they are not general statements, nor do they involve any unknowns. And the letters that appear in the 'rules' don't stand for numbers. Instead, what we have here is the equivalent of fruit salad algebra, or what I have dubbed elsewhere as letter as object.
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THURSDAY: Here we look at a task from an earlier edition of Wednesday's UK textbook series. The exercise is designed purely as a device for practising algebraic manipulation. But what does it convey to students about the purpose and utility of algebra - or, indeed, of geometry?
It turns out that, treated with any kind of common sense, shape e collapses into nothing. Curiously, exercises of this sort, which reduce algebra to an exercise in manipulation, and which abuse geometry by treating it merely as a means to this end, are commonplace in UK textbooks.
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FRIDAY: Finally, we consider a task from a recent UK adaptation of a Singapore textbook, which describes itself as The Mastery Course for Key Stage 3.
It helps if the algebra tasks we give students demonstrate the purpose and utility of algebra (to use a phrase coined by Janet Ainley and Dave Pratt). However, this is not always easy to bring about; in the case of this task, its absurd nature suggests that the exact opposite has been achieved!
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NEXT WEEK: algebra takes control and we try to make sense of the consequences...