Absolute value and the 3A MAT course

Ok. Absolute value - easy enough, to take the absolute value of a number, make it positive if it wasn't already. easy peasy...

... until you start to look at IyI=IxI and ask students to graph it..

... then ask them to find the intersection of Ix-3I=I2x+4I algebraically
... then ask them to find Ix-3I<=I2x+4I

Students really bogged down when they reached inequalities. The approach I used was similar to that by Sadler in his 3A book for MAS. The problem was that I really wasn't sure they understood what they were doing.. they could follow the algorithm but understanding was eluding them.

I started by looking at absolute numbers and explained models for solving using number lines, graphing and algebraically. Then I used a composite approach to assist students visualise what it was they were doing with problems like:

Ix-3I<=I2x+4I

I started by displaying the graph on the board using the overhead gadget for the Classpad.

I entered Graph&Tab from the menu workpane (using the menu icon at the base of the workpane).

I selected Graph&Tab and entered Ix-3I for y1 and I2x+4I for y2. For some reason the graph workpane doesn't allow you to use the 2d tab absolute value option - so use the abs() function under the cat tab in the soft keyboard. When you hit enter it will restore the absolute value notation.

Make sure both y1 and y2 are ticked (if they are not place the cursor on the line using your stylus and hit exe). Hit the graph button in the toolbar (the first icon with the top formula pane selected). The following graph should appear:

We then looked at the original inequality again and I asked what did it really mean?

Ix-3I<=I2x+4I

One way of thinking about it was, "when is the graph y=Ix-3I less than or equal to the graph of y=I2x+4I?"

We looked at the graph and found that the part marked red on the line y=Ix-3I satisfied the inequality.

Using the intersect function under analysis in the menu bar we know that the two lines intersect at -1/3 and -7 therefore the interval is x<=-7, x>=-1/3.

We had discussed that we could also do this algebraically by using the property if IxI=IyI then x=y or x=-y to find the points of intersection.

Eg.

Ix-3I<=I2x+4I

x-3 = 2x+4

-x = 7

x=-7

x-3 = -(2x+4)

x-3=-2x-4

3x=-1

x=-1/3

I then asked students to draw a number line with the intervals marked and substitute the values back into the original inequality. We numbered the three intervals. The first interval represented x<=-7, the second -7<=x<=-1/3 and the last x>=-1/3.

We then selected a value within each of the intervals and substituted them into the inequality. If they were true then this indicated values of x that satisfied the inequality.

Ix-3I<=I2x+4I

Interval 1 (x=-8)

I-8-3I<=I2(-8)+4I

I-11I<=I-12I

11<=-12 (true)

Therefore x<=-7 is a valid interval.

Interval 2 (x=-5)

I-5-3I<=I2(-5)+4I

I-8I<=I-6I

8<=-6 (false)

Therefore -7<=x<=-1/3 is not a valid interval.

Interval 3 (x=0)
I-0-3I<=I2(-0)+4I

I-3I<=I6I

3<=6 (true)

Therefore x>=-1/3 is a valid interval.

The inequality Ix-3I<=I2x+4I is valid over x<=-7, x>=-1/3

Drawing students attention from the graph and back to the algebraic representation released the tension in the room, the screwed up faces and suddenly lights went back on.

Thank goodness!

Here is a link to other CAS calculator posts.

Eureka.. one problem solved!

Teaching students to solve equations with the balancing method can be difficult as many different skills are required. Collecting like terms, fractions, multiplying pronumerals, dealing with coefficients and the like. When adding all the complexities together students can really struggle.

Surfing around yesterday I found the following link using a classpad calculator to assist students check their understanding of how to use the balancing method. It has worked fabulously well and yr10 students that typically hate algebra (and maths) are all smiles...

Here's the sequence of lessons up to this point (first week of term one)..

1. review of algebraic terminology
2. review of collecting like terms
3. review of multiplying algebraic terms
4. solving simple equations

Students were shown x + 5 =7 and asked what was a possible value for x. They responded 2 and we discussed how the observation method is often a good method for solving equations. We discussed how this was good for simple cases but with more complicated examples it became too difficult.

I then introduced the balancing method saying we could get to the same result by making x the subject of the equation by examining the LHS and thinking what operation could we do to isolate the x value.

A student suggested that we subtract 5 and I said great.

Then I said to students that the crux of the balancing method was that anything we did on the LHS of the equation has to be done to the RHS. I wrote on the board

x + 5 = 7
x + 5 -5 = 7 - 5
x = 2

and asked how did that compare with our original answer. We then did the following example:

5x + 5 = 20

A student offered the following step:
5x + 5 - 5 = 20 -5
5x = 15

Typically students get stuck at this stage as 5÷5 =1 is not an intuitive step. For once I told them that I would divide by five and showed them how it works.

5x ÷ 5 = 15 ÷ 5 (please excuse the division symbol, I actually used fractions but it is too hard in html)

x = 3

And here's the real magic.. I then took out CAS calculators borrowed from the senior school and they did a number of examples with them. For the following example:

2x - 2 = 15

Their brains started making connections and they actually were using the calculators to check that their logic was correct rather than to give them just answers.

You can see from the example that each step in the calculator mimics the steps to answer the problem on paper. It is easy to see how after each operation (+2, ÷2) x becomes the subject of the equation and ultimately becomes solved with x=8.5









Common errors become obvious earlier. Students decide what operation needs to be done and see what that operation would do. Take this common case:

The student has multiplied by 2 before they have subtracted. They can instantly see their mistake (the LHS of the equation looks more complex rather than simpler so the student starts again. The second attempt subtracting 5 gets them closer to making x the subject of the equation.

For many of us, this is how we learnt to transpose equations - a little trial and error. Lots of practice. Lots of heartache. Lots of looking at the back of the book.






The students found using the calculator fun... and the calculator only gave them guidance - not just solving the answer. It was a mix between the old inverse operations method (change the sign/change the sign that causes all sorts of difficulties when fractional terms/multiple terms are introduced) and the balancing method. To be honest, I've never found the 'scales' explanation that typically accompanies the balancing method useful - but the CAS introduction way I think may have real promise.

The other great thing is that they were recording their answers really well on paper.

For the above example I would see (with equals signs aligned):

2x-2=15
2x-2+2=15+2
2x=17
2x÷2=17÷2
x=8.5

To see mid tier students lay out work like this rather than
1) 8.5
was fantastic.

Here is a link to other CAS calculator posts.

Talented students and self confidence

If a talented student at the start of year 11 wants to leave your class what do you do?

It's a question I don't know the answer to and is a difficult one.

Paths I've taken in the past have included:
a) Discuss their choices and investigate their motivation for leaving
b) Direct them to school counsellors
c) Do nothing

Invariably before now I have been sucked into option a). This last time I've decided to do c). Whether it is peer pressure, lack of support from parents, lack of confidence in your abilities, interference from other learning areas to bolster numbers, laziness or poor work ethic; students feel compelled to make changes at the start of year 11. It will be interesting to see what they will do. I know though that I can't in all honesty tell them that they will pass if they think the grass is greener elsewhere. I'd much rather have those students that are enjoying themselves in the new courses.

The pall cast by students wishing to leave really dampens my enjoyment of classes as I was really looking forward to working positively with them. I suppose it's just the ups and downs of working with adolescents.

The main course affected seems to be the year 11 3ab MAS course. The introduction of new content seems to have spooked a few students. I am concerned that the MAT only students this year will struggle in 3C MAT next year without the additional practice provided by 3AB MAS. It is only guesswork at this stage.

Maths backgrounds for worksheets and Powerpoint

As an upper school teacher, we sometimes ignore the requirements of appearance of our materials and focus on presenting quality content. This is evident in the material available on the web.

When I get a moment I like to (in the words of my wife) make things pretty and spend the time to ensure there is a background on worksheets, a powerpoint slide with a coloured background and the like.

One site I like to use is http://www.brainybetty.com/. There are some great free backgrounds there for Powerpoint. I really like the orange one and have used it a lot.

Here is a background I made for a recent certificate for the summer school using the MSWord 2003 equation editor.

Have no fear the image is bigger. Just click on it, right click the image and save it. Here's a simple way to make it a watermark in Word 2003.

1. Open up a Word document.
2. Go Format->Background->Printed Watermark
3. Select the Picture Watermark option
4. Click Select Picture
5. Find where the background is saved and select it
6. Change the scale to 200% (or to whatever works for your image unless you want a tiled image)
7. Ensure the washout checkbox is selected
8. Click apply

Voila. A professional background for your MSWord 2003 document.

Sometimes you will find the washout selection too light for your printer. In that case you need to insert the picture, send it to behind the text and increase the brightness of the picture to a printable level that doesn't interfere with the readability of the text. This is the way I usually do it as my printer is temperamental but it makes it a bit of a cow to work with the page.

Here's how it turned out.