WEBVTT

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[Music] Welcome to Introducing 
Students to Logarithms.

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My name is Raegen Miller,
and I am an education policy analyst.

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I taught math mostly
at the high school level

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for about twelve years in both 
urban and suburban settings.

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I am offering a perspective
as an experienced teacher,

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and mostly it's about the pitfalls
that you can have when you are trying

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to teach students 
logarithmic functions.

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And we are talking
about avoiding those dangers

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and instead leveraging 
successfully student knowledge

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and their prior experiences
graphing functions.

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The history of logarithms is mostly
about logarithms as a tool

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for facilitating computations,
and this is where slide rules

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and tables played a huge role
in mathematics education,

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but is a completely obsolete 
point of view.

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And it's very easy for teachers 
to sort of go down this road,

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but it's quite a dangerous one
because it gives students the wrong

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impression of what logarithmic functions
are about because that aspect

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of logarithms is going to have nothing
to do with their experience.

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The second danger is that it's very 
easy to dwell on the properties

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of logarithms, essentially algebraic 
properties, that are important

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but can be quite a side road
that prevents students

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from getting acquainted with logarithms
in a successful way initially.

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So for example, there is a famous 
conversion formula: logcx = logbx/logbc.

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So this is a way of converting 
logarithms from one base

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to another, and it's very important,
but dwelling on this and doing exercises

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around this is not going to 
help students master the basic graph

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of logarithms.

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The third danger is that it's 
very tempting to work

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with an abstract base and do a lot 
of work initially with log b of x

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where b is some positive number.

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I recommend working with a 
concrete base first--

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something greater than one.

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The fourth danger is working
with a famous base of logarithms.

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There are a couple of famous bases.

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Base 10 is very famous base
for logarithms and base e,

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e is the base of the 
natural logarithm.

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The fifth danger is approaching
logarithms as a matter

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of inverse functions, and 
inverse notation is sort

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of notoriously difficult; it's 
a little bit counterintuitive.

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Having the inverse notation
in the same vicinity

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as a logarithmic expression, when 
students are still not comfortable

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with a logarithmic expression,
and what it means

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at all is just a 
recipe for trouble.

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Logarithmic functions are just
functions, and students have graphed

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many functions successfully, and 
we can take advantage of that fact

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to get them very quickly 
comfortable with the basic idea

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of a logarithmic functions graph.

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Let's use base two.

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So, this is log2 of x, and that 
is a function we like to graph.

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And we will use that as the 
basis for getting kids comfortable

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with graphing, getting the essential 
graph of logarithmic functions.

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The special new thing is that 
this is a whole new notation.

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They need to learn to convert 
this expression,

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the logarithmic one,
to an exponential form,

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which is equivalent.

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So, y=log2 of x is the same as 2
to the y=x,

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and that latter expression is the one 
we can work with to set up a table

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of values that we can use
to make a graph.

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And there is where we can leverage 
prior student knowledge

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because they have got a really good 
sense of habits that are involved

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with graphing a function.

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If they have the rule y=1/3x+7--
this is something they have had clearly

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before they are going to ever wind
up looking at logarithms--

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they know that if they want
to make a table of values

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to start graphing this, the thing 
to do is choose values of x

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that are convenient because the 
way we are going to work

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with this new function that's been
translated into an exponential

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expression does have a little bit
of a twist to it.

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So here again, I have y=log2 of x,
and we want to graph that,

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get the essential graph 
of that function.

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So the first step, remind them 
2 of the y=x is the

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equivalent expression.

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Another way of putting it is, "What 
is the exponent that you would put

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on 2 in order to get x?

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So, y is the exponent we would 
put on 2 in order to get x?"

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and that's kind of a formulation
to help remind them how to make

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that translation.

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Then we are going to set up a table,
and as usual we want to choose values

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of x that are convenient.

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For students to think of the values 
of x that are convenient is little bit

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of a new experience.

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If x looks like 2 to some number,
then it would be easy to pick

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out what y is.

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So, let's say x were 2 to the -2,
that would be convenient

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because that means y would be -2 or 
if x were 2 to the -1, well you compare

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that with 2 to the y, so y is -1.

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What's different here,
besides a little bit

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of a different clever step
in picking what's convenient

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and thinking about 
what's convenient,

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is working out the numbers 
before we can

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go and actually graph them
of the coordinates happens

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on different side.

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So on the table on the left 
side we have got to do well,

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"What is 2-2 and what is 2-1?"
Now, we have five nice clean points we

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can use to make a graph.

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I always label my axes, and here 
I have got the scales indicated

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adequately, and then 
graph the points.

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So, I have: 1,0; 1/2,-1; 1/4,-2; 
2,1; 4,2, and those five points are

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sufficient here in these red dots
to indicate where I should draw

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in a curve that passes
through all those points.

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And I can imagine what happens
as x gets larger,

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we come off to the right
and the graph still keeps going

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up pretty gradually,
and as x gets closer

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to the y-axis instead
of being 1/4 it's even smaller,

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then the graph's going to be coming 
down and hugging the y-axis.

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Once the students have had some success
graphing logarithmic functions,

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I would be interested in revisiting 
the specific famous bases,

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and also it makes the subject more alive
to talk about the history of logarithms.

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So I would bring it 
into the equation

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but only after the students 
have had success.

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What I have done here is to focus
on what the students know and how

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that can be useful in broaching a 
new very challenging topic.

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To learn more about Introducing 
Students to Logarithms,

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please explore the additional resources
on the Doing What Works website.