Dr Constable and the national curriculum

Dr Constable, our state education minister has been conspicuously absent from public education debates with the exception of this week when I read her reply regarding national curriculum impact on WA in The West newspaper.

It was a measured response that outlined the three years of implementation time being allowed, the need for an extended implementation (an extra year) in WA due to the variation between NSW, Vic syllabus and the current WA OBE based curriculum. She also raised issues with year 7 primary vs yr 7 high school, student entry ages in preschool/kindergarten, the lack of specialist teachers in primary and the need for training above normal 'PD' allocations requiring the sourcing of an additional budget for WA.

WA, with a smaller population and different educational requirements, will always have varied results and requirements to the eastern states. Competing with the Eastern seaboard is not statistically possible under the current measuring system.

It was encouraging to see an education minister that at least understood some of the issues faced by national curriculum and someone willing to make an attempt to avoid a head long rush into it. The challenge will be to address some of these issues and prevent these issues being swept under the table along with the children of Western Australia.

Absolute value

I spent a fair bit of time thinking about absolute value problems in the form |x+a| - |x-b| = c. Many students were struggling with visualising what these functions actually look like. What was happening when we try and solve them?

For example:

|x+5| - |x - 2| = 6

How could I display this equation graphically to give students an understanding of the underlying algebra to solve it?

I tried graphing y = |x+5| - |x - 2| and y=6 to find the intersection but was unsatisfied with the result as y = |x+5| - |x - 2| is not something easily tied to the absolute value concept or 'v' shaped absolute value graphs.

I was eventually satisfied with graphing y= |x+5| and y = |x-2| and then examining each part of the graph until I found a section of the graph that was 6 units apart.

For those wondering how to put it into a graphics calculator while exploring the concept

Go Menu -> Graph & Tab

Edit -> Clear All -> ok

at "Y1:" ->Softkeyboard->mth tab->select 'x'->type "x+5)" (it will change from abs(x+5) to |x+5|)

at "Y2:"->select 'x'->type "x-2)"

ensure that the boxes next to "Y1" and "Y2" are ticked

Now the temptation is to assume the answer is the intersection point.

but if we look at the equation |x+5| - |x - 2| = 6, it is asking "for what value of x is the value of |x+5| (the dotted line) subtract the value of |x-2| (the solid line) equal to 6". When is the gap between the two functions +6.

We can ignore values of x<= -5 as y=|x+5| is below y=|x-2| and the subtraction will only give negative values (we are looking for a gap of +6 which is a positive value).

We can also ignore values up to the intersection point as this also will only result in negative values.

The next place I looked is at x>=2 as the gap is constant and positive after this point (both functions have the same gradient).

at x=2, |x+5| is equal to 7 and |x-2| is equal to 0. |2+5| - |2-2| = 7. We can ignore values where x>=2 as the answer is not +6.

In fact the only possible solution has to lie between the intersection point (x~-1.5) and 2 and is probably closer to 2.

For y=|x+5| all values are positive between -1.5 < x < 2

For y=|x-2| all values are negative between -1.5 < x < 2

To ensure positive values for x-2 in the range -1.5< x < 2 we need to take the negative of (x-2) when solving the equation |x+5| - |x - 2| = 6.

x+5 - (-(x-2)) = 6
2x+3 = 6
x=1.5

Check answer:

|x+5| - |x - 2| = 6
Let x=1.5
|1.5+5| - |1.5-2| = 6
LHS =| 6.5| -| -.5|

= 6.5 - 0.5
= 6.0
= RHS

Viola.

It would also be interesting to explore |x+a| + |x-b| = c,  |x+a| = |x-b| and -|x+a| = c in a similar way.

Here is a link to other CAS calculator posts.

Keeping up and reducing doubt.

It's a hard ask keeping up sometimes. It's my first year teaching year 12 Calculus, Probability and Statistics. Your focus slips from your good year 12 kids, to the lower classes where your interest lies and all of us sudden you are faced with a crisis of confidence.

Are you good enough? Have you done enough? In a subject like maths, students need you to always be on the ball, or their confidence also suffers.

It's times like that you have to go back to basics. Do each exercise. Talk to a staff member that you trust. Get your confidence back. Maybe put some things aside for awhile.

I remember when I found out that I was on a pathway to take these classes, I wondered if I was up to the challenge, if my mathematics had risen back to that level. I argued that these kids needed the best teacher available. I still believe this should happen, but will fall in line with department wishes.

Maybe I have to rekindle some doubt in myself and do some real work to improve. It's a shame, because I'm really making some ground putting effort into my teaching capability with the lower classes. My masters research is teaching me a lot about myself and my teaching style - a teaching style that is much harder to work on with a good bunch of kids that will respond to a simple instruction.

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