ALG 3

In this week's set of tasks we try to get a better feel for a linear function by comparing it with a non-linear function. We see that as the independent variable changes at a steady rate, the values of the functions change at different rates, with one of them changing steadily.
The functions are set in a geometric context and we compare them geometrically (in two different ways, one of which is much more salient than the other), by expressing them symbolically, and by looking at numerical values in an ordered table.
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MONDAY: This is the root task for the week. It can be quite tricky ....

After students have drawn the shape for u = 3, you might want to show this:

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TUESDAY: Here we compare the area of our cross-shape with the area of another cross-shape:

You might want to present this on two separate slides, starting with the edited version below, so that students are not channeled too soon into answering the more structured second part. It is worth giving students plenty of time to explore the question, "Which shape has the larger area?".

After students have explored the task, they can check their hunches and confirm their insights with this Dynamic crosses movie. Some clips from the movie are shown below:
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WEDNESDAY: Here we repeat Tuesday’s task but we treat the cross-shapes as the nets of two open boxes. 

For one of the boxes, as u increases, the height of the box increases but the (square) base stays the same; for the other box, its base gets larger (in both dimensions) but the height stays the same. Thus one box changes in two dimensions simultaneously, while the other changes in only one. 

These changes are far less obvious when one just looks at the 2D nets, unless one re-configures them. For example, the first cross-shape is changing in two directions as u changes. However, if one cuts and re-joins the net like this (below), the change can be seen to occur in just one dimension.

We can do the same for the net of the other box. As u increases, the net expands in two dimensions, because of the u by u square (tinted).

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THURSDAY:This is Thursday's variant on the ALG 3 task. By now, students will, hopefully, have developed a good feel for how the areas of the two cross-shapes change with u: eg, that the first shape's area increases steadily for steady increases in u; and that the second shape's area changes more and more rapidly as u increases, and so, though its area is smaller for values like u = 2, it overtakes the other area at some point, namely when u gets past 5.

So in today's task, we represent the areas symbolically, with the aim of making these ideas about the changing areas more explicit and, conversely, giving meaning to the resulting algebraic expressions. This may also help students to get a better sense of the general form of what turn out to be linear and quadratic expressions.
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FRIDAY: Here we try to consolidate some of the burgeoning ideas by revisiting the numerical values found in Tuesday's task, and judiciously extending these with the aid of an ordered table.

 A possible next step would be to graph these sets of values - both as a way of illuminating the nature of the (linear and quadratic) relations, and as a way of giving meaning to graphs. And it would be worth relating this to the Dynamic crosses movie (with the mouse over the play/pause button).
The completed table looks like this:

 As we can see, as u increases by 1, 20u+25 increases by the same amount, whether we start from u=5 or u=10, whereas 20u+u2 increases by an increasing amount ....
We can illustrate this with a graph, such as this off-the-peg version from Excel.

The graph raises some interesting questions:

• The blue line appears to be straight. Is this really the case and what would this mean (algebraically, and in the context of the cross-shape's area)? 
• The red line seems to consist of ever-steeper line segments. Does this make sense? What if I had chosen different data-points - would I get different line-segments?? 
• What happens to the two graphs for negative values of u (algebraically, and in terms of the cross-shapes and their areas)?

It can also be illuminating to see how equivalent forms of the algebraic expressions relate to (and can be derived from) the shapes of the crosses. This is for the first cross-shape:

 

 And this is for the second cross-shape: