A survey from the perspective of a 25 year High School Science teacher on how Calculus is relevant to the high school science classroom, supported by CCSSM and NGSS.
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\title{\LARGE The Relevancy of Calculus in the Science Classroom; How CCSS/NGSS can help bridge the gap}
\subtitle{}
\author{Angela B. Vercher}
\institute{LSU Cain Center MNS Presentation }
\begin{document}
\maketitle
\begin{frame}[fragile]
\frametitle{Introduction}
\textbf{\Large Math CCSS/NGSS Connections to Calculus Fundamentals}
Key areas where Calculus concepts align with \textit{Common Core State Standards} and \textit{Next Generation Science Standards} include;
-\textit{Measurement;} standard measurements and units, scientific notation and conversions
-\textit{Exponential Growth and Decay}
-\textit{Rates and Proportional Relationships}
-\textit{Basic Algebra and Functions; }preview of Calculus terminology
-\textit{Analyzing and Interpreting Data;} using mathematics and computational thinking
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\frametitle{Rationale}
\textbf{*} Students typically view math as a necessary task to endure in earning a high school diploma, but irrelevant in the big picture of life.
\textbf{*} This mindset presents a huge roadblock to effectively teach science standards required at the 9th and 10th grade levels as many students possess low math conceptualization skills, evidenced by standardized test scores.
\textbf{*} It is crucial for teachers to find avenues to convince students that math is quite relevant in all facets of life, not just in the math classroom. The Math CCSS and NGSS, by design, strongly support such student outcomes.
%\begin{figure}
%\centering
%\includegraphics[width=11cm,height=4cm]{intro2.png}
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%\tiny
%Source: "Spacefilling Curves and Phases of the Loewner Equation", J.Lind, S.Rohde
%\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Example Problem 1}
Students are challenged to calculate the amount of money that would result from doubling a penny monthly for 1 year and then after 5 years. (Correlates to population growth predictions)
\[f(x)= 1(2)^x\] where x = the number of months the penny is being doubled.
\begin{itemize}
\item after 1 year (x=12) the amount = 4096 cents or 40.96 dollars
\item after 5 years (x=60) the amount =
\item $$1,152,921,504,607,000,000 pennies = (Quintillion cents)$$
\item or expressed as \(1.152921504607\times10^{18}\) exa-cents
\end{itemize}
%\begin{figure}
% \centering
%\includegraphics[width=8cm,height=4cm]{intro1.png}
%\caption{van Koch curve, the half-Sierpinski gasket, and the Hilbert space-filling curve}
%\end{figure}
%\tiny
%Source: "Spacefilling Curves and Phases of the Loewner Equation", J.Lind, S.Rohde
%\normalsize
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\begin{frame}[fragile]
\frametitle{The Calculus Connection}
%Let $U_{t}=\sqrt{\kappa}B_{t}$, where $\kappa \in \mathbb{R}$ and $B_{t}$ is a standard Brownian motion. Then, with probability one, the set $H_{t}$ is generated by a curve, i.e $H_{t}=\mathbb{H}\backslash \gamma(0,t]$ for some continuous curve $\gamma$. This $\gamma$ is called an SLE-curve. Two interesting facts about the SLE-curve:
\begin{itemize}
\item {\large \textbf{*} Students are exposed to the concept of positive infinity in this example.}
\item {\large \textbf{*} Learning to equate logarithmic values used in scientific notation to specific measurement prefixes will familiarize students with the skill of making a "u-substitution", as required in higher level math classes.}
%\begin{itemize}
%\item For $0 \leq \kappa \leq 4$ the trace $\gamma$ is simple with probability one.
%\item For $4 \leq \kappa < 8$ the trace $\gamma$ intersects itself and every point is contained in a loop but the curve is not space-filling (with probability 1).
%\item For $\kappa \geq 8$ the trace $\gamma$ is space-filling (with probability 1).
%\end{itemize}
\end{itemize}
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\frametitle{Example Problem 2}
%$\kappa=2$, yielding a simple trace:
{\large Students are asked to find the amount of radioactive Pb-212 remaining in a 64g sample after 20 days if 1 half-life is 10.5 hours.}
\begin{figure}
\centering
\includegraphics[width=8cm,height=4cm]{images/Chart10.png}
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%\small
%Source: http://iopscience.iop.org
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{The Calculus Connection}
\begin{itemize}
\item Using the formula;
\[Mass=\frac{Original Mass}{2^n} = \frac{64g}{2^{45.7}}\] where n = the number of half lives expired
\item This gives a mass of 0.000000000001119719321g that remains radioactive or
\item $$1.119719321\times 10^{-12}g$$ or $$1.119719321 picograms$$
\item In this example students can visualize and ponder the idea of \textbf{\textit{negative infinity}} as well as a great time to introduce the term \textbf{\textit{Asymptote}}, which allows for better understanding of the calculus concept of a limit.
\end{itemize}
%$\kappa=6$, yielding a non-simple trace:
\begin{figure}
\centering
%\includegraphics[width=8cm,height=4cm]{nonsimple.png}
\end{figure}
\small
%Source: http://iopscience.iop.org
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Example Problem 3}
%Rougher grid left, finer grid right
\begin{figure}
{\large Students perform an activity called \textbf{\textit{}}Calculating Pi to practice indirect measurement using circular objects of various sizes, then are asked to construct a graph of their data in terms of}
\textbf{\textit{Circumference}} with respect to \textbf{\textit{Diameter}}
\centering
\includegraphics[width=8cm,height=4cm]{images/Chart9.png}
\end{figure}
%\small
%Source: "Finding SLE paths in the Gaussian free field ", S.L. Watson
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{The Calculus Connection}
\begin{itemize}
\item{Students must determine independent and dependent variables and appropriately label them on the axes using appropriate units.}
\item{This is a good time to introduce the Calculus concept of rates of change in terms of}
\[\frac{dy}{dx}\]
\item{in other words,}
\[\frac{rise}{run}\]
\item giving us the slope (m) of the line
\item{The learning target is to have students correlate the ratio proportion of \textbf{\textit{Pi}} to the slope of the line.}
\end{itemize}
%Aim: Investigate upper bounds of the quantity $|\arg g^{-1 \prime}_{t}(z)|$ as $z$ approaches the boundary, for driving functions of H\"older-1/2 continuous driving functions\\
%(Recall H\"older-1/2 continuity: $|U_{t+s}-U_{t}| \leq \sigma \sqrt{s}$ for some $\sigma \in \mathbb{R}^{+}$)
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\begin{frame}[fragile]
\frametitle{Example Problem 4}
{\large Students are instructed to write a story which explains the motion of the moving object(s) that would result in graphs such as these.}
*a discussion about the corner points should emphasize how the lines are averages of the actual motion.
\begin{figure}
\centering
Distance vs. Time------------------Velocity vs. Time
\includegraphics[width=5cm,height=3cm]{images/Chart1.png}
\includegraphics[width=5cm,height=3cm]{images/Chart2.png}
%\begin{itemize}
%\item $\boldsymbol{\sigma<2\sqrt{2}}$: Nontrivial bound for argument should exist and this should be possible to prove with methods from RTZ\\
%\item $\boldsymbol{2\sqrt{2} \leq \sigma < 4}$. Non-trivial bound should exist for the argument since it does exist for the absolute value for $\sigma <4$; but this cannot be proved with methods outlined in RTZ
%\end{itemize}
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\frametitle{The Calculus Connection}
\begin{figure}
These Distance-Time graphs illustrate the following:
\centering
\includegraphics[width=4cm,height=2cm]{images/Chart3.png}
\includegraphics[width=4cm,height=2cm]{images/Chart4.png}
\includegraphics[width=4cm,height=2cm]{images/Chart5.png}
{\large constant velocity ////// acceleration ////// no motion}
\end{figure}
\begin{figure}
These Velocity-Time graphs illustrate the same:
\centering
\includegraphics[width=4cm,height=2cm]{images/Chart6.png}
\includegraphics[width=4cm,height=2cm]{images/Chart7.png}
\includegraphics[width=4cm,height=2cm]{images/Chart8.png}
\end{figure}
%\small
%In the article "Collisions and spirals of Loewner traces" by Lind, Marshall and Rohde, the Loewner trace and conformal map of the driving function $U_{t} = \sigma\sqrt{1-t}$, $0 < \sigma < 4$ are calculated. In fact here the following is stated:\\
%Given $0 < \sigma < 4$, set $\theta := -\sin^{-1}(\sigma/4)$, $\beta = 2ie^{i\theta}$, and set
%\begin{center}
%$k(z) = \frac{(z-\beta)(z-\bar{\beta})^{e^{2i\theta}}}{(\sigma - \beta)(\sigma - \bar{\beta})^{e^{2i\theta}}}$ \\
%$g_{t}(z) = (1-t)^{1/2}k^{-1}((1-t)^{-\cos\theta e^{i\theta}}k(z))$
%\end{center}
%Then $k$ is a conformal map of $\mathbb{H}$ onto $\mathbb{C}\backslash G$ where $ G := \{e^{te^{i\theta}} ; t \geq 0 \}$, is a logarithmic spiral in $\mathbb{C}$ beginning at $1$ and tending to $\infty$, and where $g_{t}$ satisfies the Loewner equation
%\begin{center}
%$\displaystyle{\dot g_{t} = \frac{2}{g_{t} - \sigma\sqrt{1-t}} \; g_{0} \equiv z} $
%\end{center}
%The trace $\gamma = k^{-1}(\{e^{-te^{i\theta}} ; t > 0\})$ is a curve in $\mathbb{H}$ beginning at $\sigma \in \mathbb{R}$ spiraling around $\beta \in \mathbb{H}$
\normalsize
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\frametitle{}
\begin{figure}
In this case the concepts of average rate of velocity and average rate of acceleration can be stressed in terms of the slope of the secent line formula: $$\frac{d(b)-d(a)}{b-a}$$
Also, the manner in which the shape and slope of the graphs are transformed as the units used to label the y-axis are changed is perfect for introducing how Newton developed equations that would calculate the velocity, acceleration and position of such an object in motion from a single graph. Thus, no need for the variety of graphs to analyze all aspects of the moving object, and students would be better prepared for learning to calculate the derivatives and integrals in order to fully analyze the moving body.
\end{figure}
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\frametitle{Acknowledgements and References}
Acknowledgements:
Special THANKS to Dr. James Madden and Dr. Frank Neubrander for their proven and persistent dedication to improving the educational future for both teachers and students in the field of the Natural Sciences. Also to Dr. Guillermo Ferrerya as Program Director of the MNS program.
References:
\begin{itemize}
\item{Glencoe Physical Science by Mclaughlin / Mcgraw-Hill, 2007 Edition}
\item{https://www.louisianabelieves.com/accountability/school-performance-scores}
\item{http://www.nextgenscience.org/get-to-know}
\item{http://map.mathshell.org}
%\item $\boldsymbol{\sigma<\sqrt{2}}$: A non-trivial bound is obtained in line with expectation. \\
%\item $\boldsymbol{\sqrt{2} \leq \sigma < 2\sqrt{2}}$: We show the methods of RTZ are insufficient for obtaining a non-trivial bound in this interval. However, we still suspect such a nontrivial bound to exist\\
%\item $\boldsymbol{2\sqrt{2} \leq \sigma < 4}$: We have shown no non-trivial bound can exist in this interval
\end{itemize}
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\plain{Questions?}
\end{document}