Katlyn Johns
10C
1/12/14
A picture says a thousand words. Those one-thousand words can be linked to many
things, and all those thoughts are linked to that one picture. That picture is the common theme.
First semester of sophomore year at Macomb Mathematics Science Technology Center
(MMSTC) can be viewed as a picture taken during the high school years in regards to math
content. Various descriptions of this snapshot include transformations, graphs, inverses,
reciprocals, viewing window, and much more. All of these descriptions fit into the shared theme
of functions. However, an even more important description could be added to the “thousand
words”. David Perkins once said, “Learning is a consequence of thinking.” A lot of thought can
be put into this small quote, especially as how it pertains to the topics of first semester.
An important aspect of learning is going back to previously attained knowledge. How can
one learn if they do not learn from mistakes, from advice, from the foundation? One must know
why something works before they can move on the content, and MMSTC achieves this by
constantly going back to previously learned content. Topics learned in the tenth grade math class
Functions, Statistics, and Trigonometry (FST) could not be understood by students had they not
learned previous topics in Geometry and Algebra with Transformations (GAT) in ninth grade.
Exponents, the complex number system, completing the square, matrices, and
introductions to right triangle trigonometry are just a few of the ninth grade topics come back to
help in FST. Knowing the laws applied to exponents helps greatly when learning logarithms
because logarithms are exponents. The product, quotient, and power properties still apply to
logarithms.
𝑙𝑜𝑔𝑏 (𝑥 ∙ 𝑦) = 𝑙𝑜𝑔𝑏 𝑥 + 𝑙𝑜𝑔𝑏 𝑦
𝑥
𝑙𝑜𝑔𝑏 ( ) = 𝑙𝑜𝑔𝑏 𝑥 − 𝑙𝑜𝑔𝑏 𝑦
𝑦
𝑙𝑜𝑔𝑏 (𝑥 𝑦 ) = 𝑦𝑙𝑜𝑔𝑏 𝑥
Product Property
Quotient Property
Power Property
Katlyn Johns
10C
1/12/14
The complex number system is also useful in naming the domain and range of functions being
learned. Completing the square, associated with the quadratic equation and formula, can be used
to solve a new model, the falling bodies model. The falling bodies model, associated with
physics, can be used with a quadratic model to find when objects hit the ground at a certain
velocity. Matrices are used to find a non-regression model for a quadratic equation. Three points
are used to make three hypothetical equations and this
is inserted into a three by three matrix. The inverse of
this matrix is multiplied by the outcome values of the
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three points to find the a, b, and c value of the equation.
Learning the meaning of each of the trigonometric
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functions is useful in learning the reciprocal
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trigonometry functions. Both the original and reciprocal trigonometry functions are based off a
right triangle. At right are the values of the original functions. These values are then reversed for
the reciprocal functions.
Another aspect of thinking is allowing one to discover patterns. A common pattern in this
stage of math is the unit circle. Just knowing the first quadrant can
allow the values of the other major points to be found. This is because
the most commonly used radian and degree measures are at multiples
of 30°, 45°, and 60°. These values are all dividends of 360° (one
revolution) so their counterparts can be found around the circle.
Going back to the idea of revisiting previously learned content, this can also be applied to
connecting math concepts to other topics. An exponential function, for example, can be used to
study continually rising populations, even though in nature, this is not a likely model. As
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Katlyn Johns
10C
1/12/14
mentioned previously, the falling bodies model, discovered by Sir Isaac Newton, is used to
express the relationship between acceleration due to gravity, velocity, and initial height of the
object. Had he not thought about this connection, this would be an unknown fact that is plainly
accepted as true. Dimensional analysis, another connection between math and science, was
originally learned in chemistry as a way to convert between different units. Later on, this concept
was used in math to convert between degrees and radians. They are related in that they both use
conversion factors, again the work of one’s thoughts.
In the process of learning new parent functions, particularly more challenging ones, the
key is knowing their qualities. Functions each have their own domain, range, and passing
coordinates, and knowing why they are what they are makes them all the easier to graph.
Knowing why their parts act as they do is the first step in learning them. For example, the inverse
trigonometry functions are restricted and pass through specific points. They are restricted
because, otherwise, they would not be functions. An inverse graph of a function is all the original
data points, but the input and output are switched. A function cannot have more than one output
for each input, and when sine, cosine, or tangent is inverted, this is not true. That is why the
inverses of the trigonometry functions are restricted to where there is only one output for each
input. One cannot learn a new function and what it looks like until they can completely
understand why it looks the way it does. Not knowing its qualities such as domain, range, and
passing points means struggling.
“Learning is a consequence of thinking”. So much power and thought come from this
quote. It speaks out as one cannot learn without thinking, therefore we cannot learn something
new unless we dig deep. All that has been learned in the first semester of FST has come from
thinking, whether it has been making connections to other topics, discovering patterns, or
Katlyn Johns
10C
1/12/14
revisiting what we already know. These aspects have impacted all the students of MMSTC to
learn what is taught and whatever is thrown their way.
Katlyn Johns
10C
1/12/14
Bibliography:
Instruments, Texas. "TI-Nspire." Texas Instruments, 2006-2013.