ALG 5

In this week's set of ALG 5 tasks, we find specific unknown values by using the so-called bar model (curved bars in our case...) and by forming and equating algebraic expressions - and we look at alternative ways of forming the expressions.
MONDAY: The root ALG 5 task. A sketch might help or you might find another way to visualise or represent the situation - or you could use trial and improvement. You can view a dynamic version of the queue here.

Note: there's plenty of scope for varying the task.
Here's an easy variant:
When has Deka queued for 11 times as long as Eric?
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Note 2:  The bar model (or in this case, 'bent rod model') is a powerful device for visualising quantities, but what if you decide to make a sketch and the proportions in your drawing turn out to be not that good? That's probably OK, as long as you don't expect instant answers from the model and are prepared to slow down and modify the drawing (on paper or in your head).

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TUESDAY: Here we ask for a symbolic algebra approach. Constructing the expressions can be quite challenging, though we can check to see whether they make sense by making use of the values found on Monday, ie by checking to see whether we get a value of 30 when e = 10. Then, having got the expressions, we can consider how they could themselves be used to derive the value e = 10, eg by forming an equation and solving it in some way ....

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WEDNESDAY: In this variant, we solve the same problem by forming expressions in d, the time taken by Deka, instead of e, the time taken by Eric.

As well as solving the problem itself (whose solution we already know, of course), this gives us the opportunity to compare the two sets of expressions and to consider how an expression for e in terms of d is related to, and can be transformed into, the corresponding expression for d in terms of e.
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THURSDAY: Here we use algebra to solve a slightly more complex problem which isn't quite as amenable to a bar-model approach as the previous task. So where previously we tried to give meaning to an algebraic approach, here we try to show its utility.

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FRIDAY: Our final version of the ALG 5 task involves the same context but has a different structure. We again represent/solve it using the clock-diagram (or bar model) and in symbolic algebra - which approach do you find easier here?

We can also approach the task using a mix of trial and improvement and analysis: 

At present (with d = 20 and e = 10) Deka and Eric have queued for a total of 30 minutes. If we wind forward 5 minutes, say, this will add 10 minutes to the total, so if we wind forwards a total of 15 minutes (making d = 20+15 and e = 10+15) they will have queued for a total of 60 minutes. This movie might help students see when to 'stop' the clock: Long lunch with Deka and Eric.

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Another approach would be to argue like this:

30+30 = 60

31+29 = 60

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35+25 = 60.

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These approaches are not purely empirical - they make use of the task's structure so they can be classed as algebraic, I think. But they don't make use of symbolic algebra, even though the last method could be said to embody d+e=30 and d–e = 10. This highlights a real dilemma: it is not that easy to devise accessible tasks where the need for symbolic algebra is compelling. However, as we shall see in later weeks, one strategy is to make the task more complex, so that symbolising provides a way of keeping track of the information.

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Notice: There's an interesting symmetry in the clock-diagram when d + e = 60: