\documentclass[12pt]{article}\usepackage{ammaths}\en

\begin{document}

\module{Vectors with Geogebra}{1}{13}{1 period}

%\prereq{}

\object{\begin{itemize}
        \item Discover the concepts of translation and vectors.
	\end{itemize}}

\mater{\begin{itemize}
	\item Task sheet
	\end{itemize}}


\modpart{Computer activity}{Whole period}

Students are working alone on a computer. They have to complete some tasks using GeoGebra, to introduce the concepts of
translation and vectors.


\pagebreak

\moddocdis{Vectors with Geogebra}{1}{13}{Task sheet}

In this session, you will work with GeoGebra on two new concepts, \emph{translation} and \emph{vector}. All answers can
be
written on this paper.

First, open Geogebra and translate it into English. Then, in the View Menu, put the Axes off and the Grid on. We will
not use coordinates in this activity.

\sectionblack{Task \#1}

\begin{enumerate}
\item Place 5 points $A$, $B$, $C$, $D$, $E$, so that no three of them are collinear.
\item \begin{enumerate}
	\item Build the midpoint $I$ of segment $BC$. 
	\item Use the ``Reflect Object in Point'' to draw the point $C'$ such that $I$ is also the midpoint of
$AC'$. 
	\item What can you say about the quadrilateral $ABC'C$ ? Prove it.\\
	.\dotfill\rule{0pt}{20pt}\\
	.\dotfill\rule{0pt}{20pt}\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item \begin{enumerate}
	\item Build the midpoint $J$ of segment $BD$. 
	\item Build the point $D'$ such that $J$ is also the midpoint of
$AD'$. 
	\item What can you say about the quadrilateral $ABD'D$ ?\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item \begin{enumerate}
	\item Build the midpoint $K$ of segment $BE$. 
	\item Build the point $E'$ such that $K$ is also the midpoint of $AE'$. 
	\item What can you say about the quadrilateral $ABE'E$ ?\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item \begin{enumerate}
	\item In the ``Line through Two Points'' menu, choose the tool ``Vector between Two Points''. Use it to draw the vectors
$\vect{AB}$, $\vect{CC'}$,  $\vect{DD'}$ and  $\vect{EE'}$.
	\item What can you say about these vectors ?\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item Use the mouse to move the point $C$.
	\begin{enumerate}
	\item What do you notice about points $I$ and $C'$ when you do so ?\\
	.\dotfill\rule{0pt}{20pt}
	\item What do you notice about vector $\vect{CC'}$ ?\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\end{enumerate}

In this situation, we say that the points $C'$, $D'$ and $E'$ are the images of $C$, $D$ and $E$ under the translation
of vector $\vect{AB}$.

\begin{flushright}
\fbox{Call the teacher to check the answers before proceeding to task \#2.}
\end{flushright}


\pagebreak

\sectionblack{Task \#2}

Open a new Geogebra document, with the axes off and the grid on.

\begin{enumerate}
\item Place two points $A$ and $B$, not on the same horizontal line but not too far apart, and draw the vector
$\vect{AB}$.
\item	\begin{enumerate}
	\item Draw a triangle $CDE$ with area $6$.
	\item Use the method introduced in task \#1 to draw the images $C'$, $D'$, $E'$ of $C$, $D$, $E$ under the
	translation of vector $\vect{AB}$.
	\item Draw the triangle $C'D'E'$.
	\item What can you say about the triangles $CDE$ and $C'D'E'$ ? What is the area of $C'D'E'$ ?\\
	.\dotfill\rule{0pt}{20pt}\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item	\begin{enumerate}
	\item Draw a square $FGHI$ with side $3$.
	\item Use the tool ``Translate Object by Vector'' to draw the images $F'$, $G'$, $H'$, $I'$ of
$F$, $G$, $H$, $I$ under the translation of vector $\vect{AB}$.
	\item Draw the quadrilateral $F'G'H'I'$.
	\item What can you say about the quadrilateral $F'G'H'I'$.\\
	.\dotfill\rule{0pt}{20pt}\\
	.\dotfill\rule{0pt}{20pt}
	\end{enumerate}
\item Use the mouse to move the point $B$. What's going on when you do so ?\\
	.\dotfill\rule{0pt}{20pt}\\
	.\dotfill\rule{0pt}{20pt}
\end{enumerate}

\begin{flushright}
\fbox{Call the teacher to check the answers before proceeding to task \#3.}
\end{flushright}

\sectionblack{Task \#3}

Open a new Geogebra document, with the axes \emph{and} the grid off.

\begin{enumerate}
\item Place three non-collinear points $A$, $B$, $C$, then draw the vectors $\vect{AB}$ and $\vect{BC}$.
\item Draw a regular pentagon $P$ away from $A$, $B$ and $C$.
\item Use the ``Translate Object by Vector'' to draw with just two clicks the image $P'$ of the pentagon $P$ under the
translation of vector $\vect{AB}$.
\item Use the same method to draw the the image $P''$ of the pentagon $P'$ under the translation of vector $\vect{BC}$.
\item What transformation maps $P$ to $P''$ ? Check your answer by applying this transformation.\\
	.\dotfill\rule{0pt}{20pt}
\end{enumerate}

\begin{flushright}
\fbox{Call the teacher to check the answer.}
\end{flushright}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%CORRECTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagebreak

\moddocdis{Vectors with Geogebra}{1}{13}{Correction sheet}

In this session, you will work with GeoGebra on two new concepts, \emph{translation} and \emph{vector}. All answers can
be
written on this paper.

First, open Geogebra and translate it into English. Then, in the View Menu, put the Axes off and the Grid on. We will
not use coordinates in this activity.

\sectionblack{Task \#1}

\begin{enumerate}
\item Place 5 points $A$, $B$, $C$, $D$, $E$, so that no three of them are collinear.
\item \begin{enumerate}
	\item Build the midpoint $I$ of segment $BC$. 
	\item Use the ``Reflect Object in Point'' to draw the point $C'$ such that $I$ is also the midpoint of
$AC'$. 
	\item What can you say about the quadrilateral $ABC'C$ ? Prove it.\\
	\textcolor{magenta}{$ABCC'$ is a parallelogram because its diagonals have the same midpoint.}
	\end{enumerate}
\item \begin{enumerate}
	\item Build the midpoint $J$ of segment $BD$. 
	\item Build the point $D'$ such that $J$ is also the midpoint of
$AD'$. 
	\item What can you say about the quadrilateral $ABD'D$ ?\\
	\textcolor{magenta}{$ABDD'$ is a parallelogram too for the same reason.}
	\end{enumerate}
\item \begin{enumerate}
	\item Build the midpoint $K$ of segment $BE$. 
	\item Build the point $E'$ such that $K$ is also the midpoint of $AE'$. 
	\item What can you say about the quadrilateral $ABE'E$ ?\\
	\textcolor{magenta}{$ABEE'$ is a parallelogram  too for the same reason.}
	\end{enumerate}
\item \begin{enumerate}
	\item In the ``Line through Two Points'' menu, choose the tool ``Vector between Two Points''. Use it to draw the vectors
$\vect{AB}$, $\vect{CC'}$,  $\vect{DD'}$ and  $\vect{EE'}$.
	\item What can you say about these vectors ?\\
	\textcolor{magenta}{They are equal because all of them have the same length and the same direction than $\vect{AB}$.} 
	\end{enumerate}
\item Use the mouse to move the point $C$.
	\begin{enumerate}
	\item What do you notice about points $I$ and $C'$ when you do so ?\\
	\textcolor{magenta}{They move.}
	\item What do you notice about vector $\vect{CC'}$ ?\\
	\textcolor{magenta}{It move but don't change.}
	\end{enumerate}
\end{enumerate}

In this situation, we say that the points $C'$, $D'$ and $E'$ are the images of $C$, $D$ and $E$ under the translation
of vector $\vect{AB}$.

\begin{flushright}
\fbox{Call the teacher to check the answers before proceeding to task \#2.}
\end{flushright}


\pagebreak

\sectionblack{Task \#2}

Open a new Geogebra document, with the axes off and the grid on.

\begin{enumerate}
\item Place two points $A$ and $B$, not on the same horizontal line but not too far apart, and draw the vector
$\vect{AB}$.
\item	\begin{enumerate}
	\item Draw a triangle $CDE$ with area $6$.
	\item Use the method introduced in task \#1 to draw the images $C'$, $D'$, $E'$ of $C$, $D$, $E$ under the
	translation of vector $\vect{AB}$.
	\item Draw the triangle $C'D'E'$.
	\item What can you say about the triangles $CDE$ and $C'D'E'$ ? What is the area of $C'D'E'$ ?\\
	\textcolor{magenta}{$\triangle CDE$ and $\triangle C'D'E'$ have the same shape and the same area.}
	\end{enumerate}
\item	\begin{enumerate}
	\item Draw a square $FGHI$ with side $3$.
	\item Use the tool ``Translate Object by Vector'' to draw the images $F'$, $G'$, $H'$, $I'$ of
$F$, $G$, $H$, $I$ under the translation of vector $\vect{AB}$.
	\item Draw the quadrilateral $F'G'H'I'$.
	\item What can you say about the quadrilateral $F'G'H'I'$.\\
	\textcolor{magenta}{The quadrilateral $F'G'H'I'$ is also a square with side $3$.}
	\end{enumerate}
\item Use the mouse to move the point $B$. What's going on when you do so ?\\
	\textcolor{magenta}{The vector $\vect{AB}$ change so the square $F'G'H'I'$ move according to the move of $B$.}
\end{enumerate}

\begin{flushright}
\fbox{Call the teacher to check the answers before proceeding to task \#3.}
\end{flushright}

\sectionblack{Task \#3}

Open a new Geogebra document, with the axes \emph{and} the grid off.

\begin{enumerate}
\item Place three non-collinear points $A$, $B$, $C$, then draw the vectors $\vect{AB}$ and $\vect{BC}$.
\item Draw a regular pentagon $P$ away from $A$, $B$ and $C$.
\item Use the ``Translate Object by Vector'' to draw with just two clicks the image $P'$ of the pentagon $P$ under the
translation of vector $\vect{AB}$.
\item Use the same method to draw the the image $P''$ of the pentagon $P'$ under the translation of vector $\vect{BC}$.
\item What transformation maps $P$ to $P''$ ? Check your answer by applying this transformation.\\
      \textcolor{magenta}{$P''$ is the image of $P$ under the translation of vector $\vect{BC}$.}
\end{enumerate}

\begin{flushright}
\fbox{Call the teacher to check the answer.}
\end{flushright}

\end{document}
