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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Limited trial of National Curriculum.

Again, on a rushed timetable, the government pushes out information that a trial is to be done this year (5 weeks into term). Teachers will be given programmes (at the 11th hour) and the kids will have to deal with a poorly understood curriculum by teachers through no fault of the teachers themselves.

Successful project management is not rushed and has an understanding of as many factors as possible. Head-in-the-sand management is a recipe for disaster. Success becomes a factor of luck rather than good management. Children's futures should not be a part of a recipe for the re-election of the Labor party at the next election. It should be a bipartisan agreement implemented with long term planning and proven methods.

Regardless of any issues with the trial, the national curriculum will be rolled out next year. What is of bigger concern is that senior school curriculum will be rolled out later this year. I really hope senior school curriculum will be given more consideration than the lower school programme as the consequences for university entrance and TAFE integration are far more severe than upper school teachers coping with students who have suffered a partial implementation with gaps in learning.

Theory and practical application are two completely different beasts. To quote that 900 people have been involved in the theoretical design of the curriculum (with little coalface application) is not going to impress. Are these the same 900 people that designed and implemented OBE in WA? I really hope not.

The media release is found here.

Politicians... What a zoo!

I wrote to the zoo to send me a politician and they sent me a

.. Julia Gillard

.. but she is an idiot, I sent her back.


so they sent me a

.. Kevin Rudd

.. but he kept apologising in Chinese, when I finally caught him in Australia, I sent him back.


so they sent me a

.. Wayne Swan

.. but he spent the postage and gifted my savings to inflation, I sent him back.


so they sent me a

.. Brendon Abbott

.. but he was too busy kissing babies, I sent him back.


so they sent me a

.. Penny Wong

.. but who cares about climate change anymore? I sent her back.


so they sent me a

.. Peter Garrett

.. but he set my house on fire and speared the whales, I sent him back.


so they sent me a

.. Wilson Tuckey

.. I tried to send him back, but he was returned to sender.


so the zoo thought really hard and sent me a banana.

.. it fit right in at parliament house.

Julia Gillard and the national curriculum

Yes, schools can change their whole curriculum focus, understand, resource and ensure that assessment is in place for a draft curriculum that will change five times before the end of the 2010. We obviously have learnt very little from the OBE implementation fiasco.

Dear, oh dear. I hope no-one buys her "it'll be all right mate" routine.

Here comes another round of teacher bashing when poor direction from government is the issue. I heard Kevin Rudd accept personal responsibility for the performance of his government. I hope he is willing to take the legal liability for rushing something through that affects so many.

Julia Gillard is again doing something in a political timeframe not appropriate to schools. Again, the children of Australia will suffer the consequences.

Where is the testing and ensuring that it is applicable in states where it is to be implemented? The issues will only become apparent under application, it needs a limited application/trial before rollout. Cynically, this won't be done due to the poor polling results of the Labor party and political necessity rather than good practice.

The sheer arrogance of the rush approach is astounding.

Manual Subtraction

An interesting question was posed to me today.

How do I subtract two number manually when the answer is negative??

For instance, 3896 - 4321 (to which the answer is -425).

I originally set up the problem in vertical columns

3896
4321
------

and tried to subtract..

3896
4321
------
?575

which obviously does not work.

So I thought about it.. the only obvious solution was to say, when subtracting always put the larger number on top.

4321
3896
------
*425

As this answer is positive, it is still incorrect. It requires an additional rule, that when the order is changed, the sign of the answer is negative. Thus the answer is -425.

I'm sure everyone knows this (and it's just one of those odd cases I haven't come across before), but it could be an interesting short investigation for upper primary or lower secondary doing directed number exercises.

.. and I have a stupid cold, my nose is dripping like a tap and I can't hug my daughter. It's made my day!

Evidence based education vs OBE

Educational trends tend to go in cycles. From ultra conservative, tried and true methods (such as direct instruction from defined syllabus) to ultra experimental (such as the whole of language approach).

Recovering from the ultra experimental 'OBE' we are now heading towards the ultra conservative 'evidence based' approach.

Although the evidence based approach has merits and is a very attractive alternative after OBE, I would suggest caution. The consequences of evidence based education is already starting to slow educational change through the inability of educational practices to change in time with social change (by the time evidence is gathered, social change has again occurred).

Current practice would be to identify an educational need, and then find a current practice (with evidence) to use to fulfil this need. The obvious issue with this is that where we have a new social situation, no evidence exists and with current research practices - no evidence will ever exist as typically research today does not seek to find a solution, only observe existing practice (existing practice which we know is flawed or wouldn't require research).

Has the pendulum swung too far, now stifling the innovative approaches that could be researched and widely implemented? To avoid this I think a middle ground needs to be found, where innovative practices are encouraged and then researched before extensive implementation. To have one without the other is to invite poor practices or stifling of positive change.