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Re: MLCS Webinar on April 24: Registration Open
Posted:
Apr 11, 2012 10:11 PM
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What if some folks don't buy that definition?
Guy
On Apr 11, 2012, at 9:55 PM, Alain Schremmer <schremmer.alain@gmail.com> wrote:
> (1) Definition: A function f is linear if and only if
>
> f( m_1?x_1 + m_2?x_2) = m_1f(x_1)+m_2?f(x_2)
>
> (The output of a linear combination of inputs is the linear combination of the outputs. In other words, the diagram commutes,)
>
> (2) The function x ???> f(x) = ax+b is NOT linear (It is called AFFINE)
>
> (3) The function x ???> f(x) = ax IS linear
>
> Regards
> --schremmer
>
>
> On Apr 10, 2012, at 4:08 PM, Philip Mahler wrote:
>
>> Re: MLCS Webinar on April 24: Registration Open
>> I wasn?t suggesting we drop connecting, or trying to connect, algebra to the real world by simplifying things. All modeling does that, and modeling is one of the most important things we can offer to our students.
>>
>> If you saw hesitancy, and you correctly did, I sometimes think of this example I saw in a textbook, problem 25, verbatim:
>>
>>> Estelle runs a television repair shop. The formula she uses to approximate her weekly profits is
>>> y = ?3x^2 + 108x
>>> where y is the profit from repairing x television sets. Use a graph to determine how many sets she must repair to make a maximum profit. What is the maximum profit?
>>
>> We do try too hard sometimes. (and I hope no lister wrote it, if so my apology, I have had colleagues that see nothing wrong with it so maybe it?s me)
>>
>> I had to go to an old talk of mine to find that, and while I did, I found this, which is a gem (albeit perhaps 10 years old and just offered because it deserves sharing):
>>
>> A young student, when asked how she decided which arithmetic technique to use in solving a problem, responded:
>> ? 3 numbers or more, I add...
>> ? 2 numbers about the same size, I subtract...
>> ? 2 numbers?one big, one little?I divide...
>> ? otherwise I multiply.
>> From an advertisement by Academic Press of books by Charles P. McKeague.
>>
>> I definitely do think that building on a real-world example that students understand well (that?s important) is a good way to approach a topic. Maybe even using just-in-time content presentation along with it.
>>
>> Phil
>>
>>
>> On 4/10/12 10:36 AM, "Blustein, Bonnie" <BlusteB@wlac.edu> wrote:
>>
>>> I assume linearity to begin with (until we have a graph and linear function that satisfies everyone) and then (usually) mention the step aspect (qualitatively, without much detail). A lot of them get the idea and appreciate that math does have a way to make an even more realistic model.
>>>
>>>
>>> From: owner-mathedcc@mathforum.org [mailto:owner-mathedcc@mathforum.org] On Behalf Of Ed Laughbaum
>>> Sent: Tuesday, April 10, 2012 6:10 AM
>>> Cc: mathedcc
>>> Subject: Re: MLCS Webinar on April 24: Registration Open
>>>
>>> I would assume that because of the mathematics level and the mathematical maturity of developmental students, it is a common classroom practice to not address the problem domain and range. For the same reason, we usually assume continuous functions when modeling real-world situations at the remedial level. Should we stop connecting remedial algebra to the real world so we can avoid issues like these?
>>>
>>> Ed
>>> =======================
>>> On 4/10/2012 7:28 AM, Philip Mahler wrote:
>>> On 4/9/12 8:55 PM, "Blustein, Bonnie" <BlusteB@wlac.edu> <mailto:BlusteB@wlac.edu> wrote:
>>> usage charge functions are usually step functions, actually. They round up to the nearest full unit.
>>>
>>>
>>> Agreed.
>>>
>>> In fact there are probably taxes added on that are a (possibly step) function of only parts of a bill. And fuel charges if it?s a car rental or plane flight... Proportions, even linear (even non-linear) function modeling can only go so far, but it is still useful.
>>>
>>> I?m trying to think of anything in a math text that is an accurate model of a real-world situation. (not to say some of those applications should not be there)
>>>
>>> One message I get from this is the importance of numeracy, which needs to include at least simple descriptive statistics. And an ability to read messy documents that have lots of data on them (like an electric bill).
>>>
>>> Still, as many others have noted since Isaac Newton, it is truly wonderous that the universe behaves in ways that can be modeled with mathematics. Not artificial situations like bills, but the very laws of physics.
>>>
>>> Phil
>>>
>>
>
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