\documentclass[12pt]{article}\usepackage{ammaths}\begin{document}

\en


\module{Using Geogebra to solve a problem}{1}{AP06}{1 period}

\prereq{None.}

\object{\begin{itemize}
        \item Using geogebra to conjuecture and solve a problem.
	\end{itemize}}

\mater{\begin{itemize}
	\item Computer room.
	\item Exercise sheets.
	\end{itemize}}

\modpart{Part A}{20 mins}

Using geogebra and working by pairs, Students have to draw a figure and to conjecture the maximal value of an
area.

\modpart{Part B}{20 mins}

Students have to find out the function involved in the problem and then to draw it (using geogebra) in order to
find the maximal value of the area.

\modpart{Part C}{20 mins}

In this part, Students have to confirm their previous answers, this time using computations.\\[10pt]

At the end of the session, the answer sheets are collected by the teacher to be marked.


\pagebreak




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First version of answers sheet with blanks to fill out
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{tabularx}{\linewidth}{|l|X|l|X|}
\hline
Student 1 & & Student 2 & \rule{0pt}{15pt} \\[5pt]
\hline
\end{tabularx}

\vspace*{1.5cm}


\moddocdis{Using Geogebra to solve a problem}{1}{AP06}{ICT}


In the figure below, $ABC$ is a right-angled isosceles triangle in $A$ such that $BC=8$ cm. $MNPQ$ is a rectangle
inscribed in $\triangle ABC$. The length $BM$ is variable, we denote it $x$, in cm.\\
The aim of this session is to find the position of the point $M$ that maximizes the area of the rectangle $MNPQ$.
\begin{center}\psset{xunit=0.7cm,yunit=0.7cm}
\begin{pspicture}(-0.5,-0.5)(5,4)
\psline(0,0)(4,4)
\psline(4,4)(8,0)
\psline(8,0)(0,0)
\psline(1.5,0)(1.5,1.5)
\psline(1.5,1.5)(6.5,1.5)
\psline(6.5,1.5)(6.5,0)
\uput[l](0,0){B}
\uput[ul](1.5,1.5){N}
\uput[u](4,4){A}
\uput[ur](6.5,1.5){P}
\uput[r](8,0){C}
\uput[d](1.5,0){M}
\uput[d](6.5,0){Q}
\psline{<->}(0,-0.3)(1.5,-0.3)
\uput[d](0.7,-0.3){$x$}
\end{pspicture}
\end{center}


\probpart{ -- Using Geogebra}
\begin{enumerate}
\item Use Geogebra to draw the figure. Don't forget to display the area of the rectangle on the figure.
\item Move the point $M$ on the segment $BC$ and conjecture the maximal value of the area of $MNPQ$.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\end{enumerate}


\probpart{ -- Using the graph of a function}

\begin{enumerate}
\item What is the minimal value for $x$ ? Explain quickly your answer.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item What is the maximal value for $x$ ? %Explain quickly your answer.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item What are the measures of the angles of $\triangle ABC$ ? of $\triangle BMN$ ? Deduce the nature of the triangle $BMN$ then the length of $MN$ as a function of $x$.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[90pt]
      \hline
      \end{tabular}
      \end{center}
      
\item Let $f$ be the function that maps any adequate value of $x$ to the corresponding area of $MNPQ$. Prove that
$$f(x)=8x-2x^2$$
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item Use Geogebra to draw the graph of $f$ and give the maximal value of $f$ and the value of $x$ for which it is reached.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item Deduce the maximal area of $MNPQ$ and the position of the point $M$ for which it's reached.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\end{enumerate}


\probpart{ -- Using computation}
In this part, let's admit that the area of $MNPQ$ is given by the expression $8x-2x^2$, where $x$ represents the length $BM$.
\begin{enumerate}
 \item What is the sign of the expression $-2(x-2)^2$ ?
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
 \item Expand the expression $-2(x-2)^2$.
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[20pt]
      \hline
      \end{tabular}
      \end{center}
 \item Using the two previous questions, deduce the maximal value of the area of $MNPQ$.
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\rule{0pt}{15pt}\\[30pt]
      \hline
      \end{tabular}
      \end{center}
\end{enumerate}

\pagebreak\raz

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solutions sheet
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\moddocdis{Using Geogebra to solve a problem}{1}{AP06}{Correction}


In the figure below, $ABC$ is a right-angled isosceles triangle in $A$ such that $BC=8$ cm. $MNPQ$ is a rectangle
inscribed in $\triangle ABC$. The length $BM$ is variable, we denote it $x$, in cm.\\
The aim of this session is to find the position of the point $M$ that maximizes the area of the rectangle $MNPQ$.
\begin{center}\psset{xunit=0.7cm,yunit=0.7cm}
\begin{pspicture}(-0.5,-0.5)(5,4)
\psline(0,0)(4,4)
\psline(4,4)(8,0)
\psline(8,0)(0,0)
\psline(1.5,0)(1.5,1.5)
\psline(1.5,1.5)(6.5,1.5)
\psline(6.5,1.5)(6.5,0)
\uput[l](0,0){B}
\uput[ul](1.5,1.5){N}
\uput[u](4,4){A}
\uput[ur](6.5,1.5){P}
\uput[r](8,0){C}
\uput[d](1.5,0){M}
\uput[d](6.5,0){Q}
\psline{<->}(0,-0.3)(1.5,-0.3)
\uput[d](0.7,-0.3){$x$}
\end{pspicture}
\end{center}


\probpart{ -- Using Geogebra}
\begin{enumerate}
\item Use Geogebra to draw the figure. Don't forget to display the area of the rectangle on the figure.
\item Move the point $M$ on the segment $BC$ and conjecture the maximal value of the area of $MNPQ$.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\textcolor{magenta}{The maximum area for $MNPQ$ seems to be $8$ cm$^2$.}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\end{enumerate}


\probpart{ -- Using the graph of a function}

\begin{enumerate}
\item What is the minimal value for $x$ ? Explain quickly your answer.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\textcolor{magenta}{The minimal value for $x$ is $0$ because $x$ represents a distance.}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item What is the maximal value for $x$ ? %Explain quickly your answer.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
	\textcolor{magenta}{The maximal value for $x$ is $4$ because $M$ can't be on the other side of the midpoints.}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item What are the measures of the angles of $\triangle ABC$ ? of $\triangle BMN$ ? Deduce the nature of the triangle $BMN$ then the length of $MN$ as a function of $x$.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{$\triangle ABC$ is a right-angled isosceles triangle in $A$ so $\angle BAC=90^o$ and $\angle ABC=\angle BCA=45^o$.}\par
      \textcolor{magenta}{In $\triangle BMN$, $\angle BMN=90^o$ because $MNPQ$ is a rectangle and $\angle MBN=\angle CBA=45^o$. So $\triangle BMN$ is a right-angled isosceles triangle in $B$.}\par
      \textcolor{magenta}{We deduce that $MN=BC=x$.}      
      \\\hline
      \end{tabular}
      \end{center}
 \pagebreak    
\item Let $f$ be the function that maps any adequate value of $x$ to the corresponding area of $MNPQ$. Prove that
$$f(x)=8x-2x^2$$
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{$f(x)=MN\times MQ=x\times(8-2x)=8x-2x^ 2$.}\\[5pt]
      \hline
      \end{tabular}
      \end{center}
\item Use Geogebra to draw the graph of $f$ and give the maximal value of $f$ and the value of $x$ for which it is reached.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{According to the graph, the maximal value of $f$ seems to be $8$ reached for $x=2$.}
      \\\hline
      \end{tabular}
      \end{center}
\item Deduce the maximal area of $MNPQ$ and the position of the point $M$ for which it's reached.
      \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{The function represents the area so the maximal value of the area seems to be $8$ cm$^2$ reached for $BM=2$ cm.}
      \\\hline
      \end{tabular}
      \end{center}
\end{enumerate}


\probpart{ -- Using computation}
In this part, let's admit that the area of $MNPQ$ is given by the expression $8x-2x^2$, where $x$ represents the length $BM$.
\begin{enumerate}
 \item What is the sign of the expression $-2(x-2)^2$ ?
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{$(x-2)^2$ is a squared thus positive. As we multiply it by $-2$ which is negative, we deduce that $-2(x-2)^2$ is negative.}
      \\ \hline
      \end{tabular}
      \end{center}
 \item Expand the expression $-2(x-2)^2$.
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{$-2(x-2)^2=-2(x^2-4x+4)=-2x^2+8x-8$.}
      \\\hline
      \end{tabular}
      \end{center}
 \item Using the two previous questions, deduce the maximal value of the area of $MNPQ$.
       \begin{center}
      \begin{tabular}{|p{14cm}|}
      \hline
      \textcolor{magenta}{According to the first question $-2(x-2)^2$ is negative so $-2(x-2)^2\leq0$.}\par
      \textcolor{magenta}{According to the second question, we get $-2x^2+8x-8\leq0$ thus $-2x^2+8x\leq8$.}\par
      \textcolor{magenta}{The last inequation means that $f(x)$ can't be greater than $8$.}
      \\\hline
      \end{tabular}
      \end{center}
\end{enumerate}


\pagebreak\raz

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Second version of answers sheet without blanks to fill out
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\moddocdis{Using Geogebra to solve a problem}{1}{AP06}{ICT}


In the figure below, $ABC$ is a right-angled isosceles triangle in $A$ such that $BC=8$ cm. $MNPQ$ is a rectangle inscribed in $\triangle ABC$. The length $BM$ is variable, we denote it $x$, in cm.\\
The aim of this session is to find the position of the point $M$ that maximizes the area of the rectangle $MNPQ$.
\begin{center}\psset{xunit=0.7cm,yunit=0.7cm}
\begin{pspicture}(-0.5,-0.5)(5,4)
\psline(0,0)(4,4)
\psline(4,4)(8,0)
\psline(8,0)(0,0)
\psline(1.5,0)(1.5,1.5)
\psline(1.5,1.5)(6.5,1.5)
\psline(6.5,1.5)(6.5,0)
\uput[l](0,0){B}
\uput[ul](1.5,1.5){N}
\uput[u](4,4){A}
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\uput[r](8,0){C}
\uput[d](1.5,0){M}
\uput[d](6.5,0){Q}
\psline{<->}(0,-0.3)(1.5,-0.3)
\uput[d](0.7,-0.3){$x$}
\end{pspicture}
\end{center}


\probpart{ -- Using Geogebra}
\begin{enumerate}
\item Use Geogebra to draw the figure. Don't forget to display the area of the rectangle on the figure.
\item Move the point $M$ on the segment $BC$ and conjecture the maximal value of the area of $MNPQ$.
\end{enumerate}


\probpart{ -- Using the graph of a function}

\begin{enumerate}
\item What is the minimal value for $x$ ? Explain quickly your answer.
\item What is the maximal value for $x$ ?
\item What are the measures of the angles of $\triangle ABC$ ? of $\triangle BMN$ ? Deduce the nature of the triangle $BMN$ then the length of $MN$ as a function of $x$.
\item Let $f$ be the function that maps any adequate value of $x$ to the corresponding area of $MNPQ$. Prove that
$$f(x)=8x-2x^2$$
\item Use Geogebra to draw the graph of $f$ and give the maximal value of $f$ and the value of $x$ for which it is reached.
\item Deduce the maximal area of $MNPQ$ and the position of the point $M$ for which it's reached.
\end{enumerate}


\probpart{ -- Using computation}
In this part, let's admit that the area of $MNPQ$ is given by the expression $8x-2x^2$, where $x$ represents the length $BM$.
\begin{enumerate}
 \item What is the sign of the expression $-2(x-2)^2$ ?
 \item Expand the expression $-2(x-2)^2$.
 \item Using the two previous questions, deduce the maximal value of the area of $MNPQ$ and the position of the point $M$ for which it's reached.
\end{enumerate}
\end{document}
