ALG 06

Week 6: Here we look at tasks involving the various ages of a group of people - in our case, Ella and her brothers and Ella and her mother. The tasks hinge on the key idea that if a person's age increases by a given amount, so does everyone else's age - the change is addititive.
We take several looks at one of the tasks and try to solve it informally, and then using a bar model, and algebraic symbols (perhaps), and with specific numbers.
-
Monday: The purpose of this task is to establish the 'rules of the game' when it comes to changes in people's age.
What happens to everyone's age when Jez, who is currently 3 years old, gets to be 12 years old?

Fitz assumes that when Jez is 4 times older than he is now (ie when he is 12, compared to 3 now) everyone will be 4 times older than they are now. In reality, they will each only be 9 years older than they are now!
If the total age changes from 25 years to 100 years, this is a total increase of 75 years for the three people, which is 25 years each.
-
Tuesday: We will revisit this task on each of the next three days, but here we leave pupils to their own devices. To solve the task, pupils need to coordinate quite a lot of information. It should be interesting to see how they do this: do they model the task in some way, eg by drawing a diagram or using specific numbers - or algebraic symbols even!

It turns out that Ella's mum will be 3 times as old as Ella when Ella's age has doubled. Notice that the multiplier here (3) is smaller than the earlier multiplier (5). This makes sense, since as Ella and her mum get older, the ratio of their ages will get closer and closer to 1!
-
Wednesday: Here we use lines or 'bars' to model the ages. By how much does Ella's age increase when it doubles?

We have represented Ella's age by a bar-segment and her mum's age by 5 such segments. This makes the situation a bit more concrete, even though we don't know what number the segment represents.
When Ella's age doubles, it increases by 1 segment, as does her mother's age. So Ella's age is now represented by 2 segments, and her mother's age by 6 segments, ie 3 times as many:

Here's an alternative version of Wednesday's task, for those who prefer a baaa model...

-
Thursday: Here we meet the same problem again, but this time we use a letter, E, to represent Ella's current age, rather than a bar-segment.

The required ages can be written as 5E, 2E and 6E, respectively. However, It is likely that some pupils will write these in less conventional ways (eg 5×E or E×5 or E5 instead of 5E) and in less simplified ways (eg 5E+E instead of 6E). You will need to decide whether to discuss these issues or to leave pupils' responses as they are if they seem to make sense.
Having established that Ella's and her mum's future ages will be 2E and 6E (or equivalent), you might want to go on to discuss how we can use this to solve the actual problem, or you might want to defer this until pupils have met Friday's version of the task.
-
Friday: Here we take a very concrete approach by choosing specific values for Ella's age. Pupils who might have struggled with the earlier unstructured or more abstract approaches should find this more accessible.

If Ella is 4 years old, her mother will be 5×4 = 20 years old. Then, when Ella reaches 8 (ie 4+4), her mother will be 20+4 = 24, which is 3 times 8.
If Ella is 10 years old, her mother will be 50, and when Ella gets to be twice as old (ie 20) her mother gets to be 50+10 = 60, which is 3 times 20, ie again 3 times as old as Ella's new age.
It is worth re-writing this kind of exposition in a more generic or 'number tracking' way, eg like this:

Ella: is 4, and later becomes 2×4 (which is 4+4)

Mum: is 5×4, and later becomes 5×4 + 4 = 6×4 = 3×(2×4).

Here, we could replace each 4 by a 10, or by another number of our choice, or by E. This should allow some pupils to perceive (or at least glimpse) the general structure of the task, which could be said to boil down to this: 5 = 5×1, (5+1) = 3×(1+1).
To get further insight into the structure of the task, you might want to change the original 'age factor'. For example, instead of mum being '5 times older' than Ella initially, you could change this to, say, '7 times older'. Do we always get a whole number 'age factor' for the solution?