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movieA: What happens to the shape of this object? (1)

The value of x changes - what happens to the shape of the object?
Pause the movie, to give you time to think.
(Note: the scale used for the drawing is not constant....)
What happens when x = 100, say? Would the object go off the screen?) bar: A dynamic bar model for an equation (1) This shows how, as the variable 'a' changes, there is a value of 'a' for which 3a + 2 = 2a + 8.
It involves the familiar and age-old bar model, but not in the usual static way. bar2: A dynamic bar model for an equation (2)
Here we show a dynamic bar model for the equation 2e + 4 = 19 – e, and further models for two transformations of the equation.
The aim is to provide insight into the algebraic relations, rather than a quick-fix solution procedure.  bar3extra: A 2D dynamic bar model for an equation (1) EXTRA
This is the same as "A 2D dynamic bar model for an equation (1)", except that initially we zoom out to allow 'a' to vary over a larger range (0 to 20, rather than 0 to 10).
Again, it matches our movie "A dynamic bar model for an equation (1)" about the equation 3a + 2 = 2a + 8.

bar3: A 2D dynamic bar model for an equation (1)
This matches our movie "A dynamic bar model for an equation (1)" about the equation 3a + 2 = 2a + 8.
What are the affordances of the 2D and 1D models?
Our sense is, the 1D model lends itself to an analytic approach, ie to transforming the given equation into something simpler.In the 2D model, it is perhaps easier to think about the relative values of the two expressions, as the value of 'a' varies ("Is the rectangle tall and thin or short and wide?"). bar4: A dynamic bar model for an equation (3)
This is the same equation as bar model movie (1), ie 3e + 2 = 2e + 8, but here the bars have morphed into rods whose lengths are the result of an enlargement.

As the variable 'e' changes, there is a value where the expressions 3e+2 and 2e+8 have the same value. bar4extra: A dynamic bar model for an equation (3) EXTRA
Here we use a dynamic bar model (or rather, hot rods) to represent the equation 3e + 2 = 2e + 8.

We then modify the scale factors used to determine the lengths of the rods and thus consider the modified equations 3e + 2 = 1.5e + 8 and 2.5e + 2 = 1.5e + 8.
The last equation turns out to have the same solution as the first. Why? bar5: A dynamic bar model for a (quadratic) equation (4a)
This bar (or rod) model movie is for the equation 36/e = e + 3.5.

It might be argued that it is pushing the model rather far to use it for such an equation, thought it is a nice challenge to make sense of the model here.
[The equation is equivalent to e^2 + 3.5e – 36 = 0. There is of course another solution, not shown.] bar5neg: A dynamic bar model for a (quadratic) equation (4b) (2 sols)
This bar (or rod) model movie is for the equation 36/e = e + 3.5. It is an 'expanded' version of movie (4a), showing negative as well as positive values for e, and thus showing the second solution e = -8.
The equation is equivalent to e^2 + 3.5e – 36 = 0 or (e+8)(e–4.5)=0. comp-square: A dynamic model of completing the square
Here we use an area model to illustrate how one could solve the equation e(e + 6) = 135 by 'completing the square'.
In effect, we are transforming the equation into (e + 3)^2 = 135 + 9 = 144. Here 'e' only appears once, so we can solve the equation relatively easily by trial-and-improvement, by 'inspection' or the 'cover-up' method, or by using further transformation rules giving e+3 = ±√ 144, etc.

[Note: we only show the positive root here. The model can be used for the other root (e = -15) but the overlapping rectangles are difficult to construe especially if both roots are to be shown on the same screen.] 