This week we look at Cartesian graphs of straight lines again. The aim is to focus on the visual aspects of the graphs, in particular the slopes of various lines, and to find points on the lines by using notions of proportion and similarity, rather than by immediately expressing the lines as equations (though that would, of course, be a perfectly legitimate thing to do).
In many of the tasks we have deliberately chosen points with very different sets of coordinates [such as (0, 0) and (100, 101) in ALG 12A] so as to encourage students to visualise or sketch what's going on, rather than attempt to use accurate drawings.

MONDAY: Many students should be able to find coordinates for some points on the red line, eg (50, 50.5) and (200, 202). Encourage students to seek out lots of points as some might provide insights that would help with parts i. and ii., eg (10, 10.1) and (1, 1.01).
It is likely that some students will suggest that the value of the ycoordinate in part i. is 102. How can we show (eg with the aid of a sketch) that a point with coordinates (101, 102) is not on the red line? How can we decide whether it is above or below?

TUESDAY: We search for some more widelyspaced points.
A sketch might help; a table of values might help, to record moves towards the yaxis, say, in wellchosen steps (eg 10 to the left, 1 up).
PS: Here's a nice solution, posted on Twitter. Quite condensed but worth unpicking!

WEDNESDAY: More points on a line, near and far...
Follow the light green straight line.... Not that different from the previous task, but no harm in consolidating earlier ideas. And you might like to think up further variations....

THURSDAY: Two lines, by way of a change.....
Here we take a somewhat different approach by offering two methods for students to interpret. The first is quite grounded, the second is more abstract. You might, of course, prefer to ask students to come up with their own methods first, though the task itself is quite demanding.

FRIDAY: By way of a further change, we home in on the microscopic  though you might want to zoom out again ....
Don't be put off by the fractions! The underlying algebraic relationships are not that difficult.... And there are also some nice geometric approaches that can be used to solve the task.
Commentary: A closer look at the coordinates reveals that for one of the line segments the coordinates add up to 1, for the other the ycoordinate is 3 times the xcoordinate. Which two numbers add up to 1, where one number is 3 times the other? We can get there fairly rapidly using trial and improvement. Or we could draw a 'bar' or 'rod' or number line (although this requires quite a difficult switch in thinking  the 'bar' is not a line segment in the Cartesian plane!):
We can also express the relations between the coordinates more formally, as the simultaneous equations x + y = 1 and y = 3x, leading to x + 3x = 1, so x = 1/4, y = 3/4.

Here's a nice approach (by Matt L) that is more geometric. It involves constructing a new point on the line through the steeper line segment that is vertically below the left hand endpoint of the other line segment. It also uses the fact that this line segment slopes at 45˚, so the two segments labelled 'a' are equal.
We can extend the two given line segments still further, until they touch a unit square. [In fact, this is what I started with when I designed the task!]
A couple of years ago there was a flurry of tasks on Twitter involving dissected squares of this sort, usually with one region shaded, the aim being to find 'What fraction of the square is shaded?' My favoured approach was to draw equally spaced parallel lines to produce equal intercepts, as here:
One could then argue along these lines: the diagonal of the square has been cut into 4 equal segments, so starting at the bottom left corner, say, of the square, point P is 1/4 units across and 3/4 units up. [Or the height of the triangle seated on the base of the square is 3/4 so its area is 3/8 of the unit square.]
A related and perhaps more direct approach (whether to locate P or to find areas) is to make use of the fact that this triangle on the base of the square is similar to the small 'upside down' triangle above it, with a base that is 3 times as long.
I remember someone coming up with an algebraic solution to one of these area tasks on Twitter. This took me by surprise and I was impressed by the power of such an approach. But it also seemed disappointing to 'abandon' the geometry! Of course, one might say the reverse about the ALG 12E task. Here an algebraic approach would seem an obvious choice and it is perhaps a bit perverse to go to the lengths of constructing a geometric approach instead!
Sunday, 29 April 2018
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
....
The extract below is from my article Object lessons in algebra? that appeared in Mathematics Teaching 98 in 1982. Still worth reading!
Here is another variant of the task, but where we are asked to write a relation rather than interpret an expression. It is again quite challenging.
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.
In a sense, this task is quite easy  we can get it right by interpreting the letters correctly, leading to 3g = b (on the basis that 3 times the length g equals the length b), but we can also get it right by writing 3g = b as shorthand for 3 green rods make a blue rod, where g and b are perhaps being thought of merely as shortened names for the green and blue rods, and not necessarily as symbols for their lengths. Thus, in a task like this, students can appear to be thinking algebraically when perhaps they're not.

FRIDAY: Here we meet the same rows of green rods and blue rods, but this time we relate the number of green rods and blue rods, not their lengths.
In this task, we can feel a strong pull towards writing 3g = b again, on the basis that 3 green rods make a blue rod, or there are 3 green rods for every blue rod. However, we need to fix firmly on the fact that in this version of the rods task, g and b are defined as numbers of rods, for example 3 and 1, or 6 and 2, or 9 and 3, etc. So g is 3 times b, ie g = 3b. This can feel counterintuitive as it doesn't map onto our verbal descriptions of the situation.
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
....
The extract below is from my article Object lessons in algebra? that appeared in Mathematics Teaching 98 in 1982. Still worth reading!
Here is another variant of the task, but where we are asked to write a relation rather than interpret an expression. It is again quite challenging.
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.

FRIDAY: Here we meet the same rows of green rods and blue rods, but this time we relate the number of green rods and blue rods, not their lengths.
In this task, we can feel a strong pull towards writing 3g = b again, on the basis that 3 green rods make a blue rod, or there are 3 green rods for every blue rod. However, we need to fix firmly on the fact that in this version of the rods task, g and b are defined as numbers of rods, for example 3 and 1, or 6 and 2, or 9 and 3, etc. So g is 3 times b, ie g = 3b. This can feel counterintuitive as it doesn't map onto our verbal descriptions of the situation.
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).

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):

TUESDAY: We look at another familiar area scenario, namely area of a trapezium (and, to keep it simple, a trapezium that is rightangled).
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 (topleft diagram, below).
The topright diagram shows what happens to the trapezium as point P moves until it is 5 units to the left of the formerly topleft 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 topleft 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 topright diagram.

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:

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 Tshape 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 Tshape?) 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 Tshape 5 squares to the right... [Are there other ways?]
Here are positions for the Tshape 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 Tshape, 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 2D grid into a continuous Cartesian plane.

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.

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.
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).

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):

TUESDAY: We look at another familiar area scenario, namely area of a trapezium (and, to keep it simple, a trapezium that is rightangled).
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 (topleft diagram, below).
The topright diagram shows what happens to the trapezium as point P moves until it is 5 units to the left of the formerly topleft 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 topleft 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 topright diagram.

WEDNESDAY: Things get really interesting.... What's a 2andahalf 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:

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 Tshape 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 Tshape?) 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 Tshape 5 squares to the right... [Are there other ways?]
Here are positions for the Tshape 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 Tshape, 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 2D grid into a continuous Cartesian plane.

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.

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 wellknown (or wellpublicised) textbooks that involve faux, fake, phony, manquÃ©, 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?

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 Tshirts. Perhaps the Singapore government has created a mechanism that discourages parttime employment...
What of her pay, I wonder. Is it per hour or per Tshirt?
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).

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.

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.

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!

NEXT WEEK: algebra takes control and we try to make sense of the consequences...
MONDAY: a peach from Singapore (or several peaches, if you can afford them).

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 Tshirts. Perhaps the Singapore government has created a mechanism that discourages parttime employment...
What of her pay, I wonder. Is it per hour or per Tshirt?
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).

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.

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.

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!

NEXT WEEK: algebra takes control and we try to make sense of the consequences...
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