Writing Formulas

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\celsius }[0]{} \newcommand {\degree }[0]{} \newcommand {\ohm }[0]{} \newcommand {\Tienc }[0]{T1} \newcommand {\oandsfamily }[0]{\usefont {OT1}{oands}{m}{n}} \newcommand {\pmhgfamily }[0]{\usefont {OT1}{pmhg}{m}{n}} \newcommand {\Hic }[0]{\cedover {\i }\ensuremath {\!}} \newcommand {\Hrp }[0]{\,\textoands {‘}} \newcommand {\Hag }[0]{\textoands {z}} \newcommand {\translitpmhgfont }[0]{\itshape } \newcommand {\newrobustcmd }[0]{} \newcommand {\csshow }[1]{\begingroup \expandafter \endgroup \expandafter \show \csname #1\endcsname } \newcommand {\csmeaning }[1]{\ifcsname #1\endcsname \expandafter \meaning \csname #1\endcsname \else \detokenize {undefined}\fi } \newcommand {\ifdefmacro }[0]{} \newcommand {\ifdefparam }[0]{} \newcommand {\ifdefprotected }[0]{} \newcommand {\ifnumequal }[1]{\ifnumcomp {#1}=} \newcommand {\ifnumgreater }[1]{\ifnumcomp {#1}>} \newcommand {\ifnumless }[1]{\ifnumcomp {#1}<} \newcommand {\ifdimequal }[1]{\ifdimcomp {#1}=} \newcommand {\ifdimgreater }[1]{\ifdimcomp {#1}>} \newcommand {\ifdimless }[1]{\ifdimcomp {#1}<} \newcommand {\expandonce }[1]{\unexpanded \expandafter {#1}} \newcommand {\csexpandonce }[1]{\expandafter \expandonce \csname #1\endcsname } \newcommand {\protecting }[0]{} \newcommand {\csdef }[1]{\expandafter \def \csname #1\endcsname } \newcommand {\csedef }[1]{\expandafter \edef \csname #1\endcsname } \newcommand {\csgdef }[1]{\expandafter \gdef \csname #1\endcsname } \newcommand {\csxdef }[1]{\expandafter \xdef \csname #1\endcsname } \newcommand {\cslet }[2]{\expandafter \let \csname #1\endcsname #2} \newcommand {\letcs }[2]{\ifcsdef {#2} {\expandafter \let \expandafter #1\csname #2\endcsname } {\undef #1}} \newcommand {\csletcs }[2]{\ifcsdef {#2} {\expandafter \let \csname #1\expandafter \endcsname \csname #2\endcsname } {\csundef {#1}}} \newcommand {\csuse }[1]{\ifcsname #1\endcsname \csname #1\expandafter \endcsname \fi } \newcommand {\appto }[2]{\ifundef {#1} {\edef #1{\unexpanded {#2}}} {\edef #1{\expandonce #1\unexpanded {#2}}}} \newcommand {\eappto }[2]{\ifundef {#1} {\edef #1{#2}} {\edef #1{\expandonce #1#2}}} \newcommand {\gappto }[2]{\ifundef {#1} {\xdef #1{\unexpanded {#2}}} {\xdef #1{\expandonce #1\unexpanded {#2}}}} \newcommand {\xappto }[2]{\ifundef {#1} {\xdef #1{#2}} {\xdef #1{\expandonce #1#2}}} \newcommand {\preto }[2]{\ifundef {#1} {\edef #1{\unexpanded {#2}}} {\edef #1{\unexpanded {#2}\expandonce #1}}} \newcommand {\epreto }[2]{\ifundef {#1} {\edef #1{#2}} {\edef #1{#2\expandonce #1}}} \newcommand {\gpreto }[2]{\ifundef {#1} {\xdef #1{\unexpanded {#2}}} {\xdef #1{\unexpanded {#2}\expandonce #1}}} \newcommand {\xpreto }[2]{\ifundef {#1} {\xdef #1{#2}} {\xdef #1{#2\expandonce #1}}} \newcommand {\csappto }[1]{\expandafter \appto \csname #1\endcsname } \newcommand {\cseappto }[1]{\expandafter \eappto \csname #1\endcsname } \newcommand {\csgappto }[1]{\expandafter \gappto \csname #1\endcsname } \newcommand {\csxappto }[1]{\expandafter \xappto \csname #1\endcsname } \newcommand {\cspreto }[1]{\expandafter \preto \csname #1\endcsname } \newcommand {\csepreto }[1]{\expandafter \epreto \csname #1\endcsname } \newcommand {\csgpreto }[1]{\expandafter \gpreto \csname #1\endcsname } \newcommand {\csxpreto }[1]{\expandafter \xpreto \csname #1\endcsname } \newcommand {\csnumdef }[1]{\expandafter \numdef \csname #1\endcsname } \newcommand {\csnumgdef }[1]{\expandafter \numgdef \csname #1\endcsname } \newcommand {\csdimdef }[1]{\expandafter \dimdef \csname #1\endcsname } \newcommand {\csdimgdef }[1]{\expandafter \dimgdef \csname #1\endcsname } \newcommand {\csgluedef }[1]{\expandafter \gluedef \csname #1\endcsname } \newcommand {\csgluegdef }[1]{\expandafter \gluegdef \csname #1\endcsname } \newcommand {\mudef }[2]{\ifundef #1{\def #1{0mu}}{}\edef #1{\the \muexpr #2}} \newcommand {\csmudef }[1]{\expandafter \mudef \csname #1\endcsname } \newcommand {\csmugdef }[1]{\expandafter \mugdef \csname #1\endcsname } \newcommand {\listbreak }[0]{} \newcommand {\listadd }[2]{\ifblank {#2}{}{\appto #1{#2|}}} \newcommand {\listgadd }[2]{\ifblank {#2}{}{\gappto #1{#2|}}} \newcommand {\listcsadd }[1]{\expandafter \listadd \csname #1\endcsname } \newcommand {\listcseadd }[1]{\expandafter \listeadd \csname #1\endcsname } \newcommand {\listcsgadd }[1]{\expandafter \listgadd \csname #1\endcsname } \newcommand {\listcsxadd }[1]{\expandafter \listxadd \csname #1\endcsname } \newcommand {\listcsremove }[1]{\expandafter \listremove \csname #1\endcsname } \newcommand {\listcsgremove }[1]{\expandafter \listgremove \csname #1\endcsname } \newcommand {\dolistloop }[0]{\forlistloop \do } \newcommand {\dolistcsloop }[0]{\forlistcsloop \do } \newcommand {\AfterPreamble }[0]{\AtBeginDocument } \newcommand {\mdfsetup }[0]{\kvsetkeys {mdf}} \newcommand {\mdfcreateextratikz }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\addvmargin }[1]{ \node [fit=(current bounding box),inner ysep=#1,inner xsep=0]{}; } \newcommand {\tri }[0]{\triangle } \newcommand {\l }[0]{\ell } \newcommand {\P }[0]{\mathcal {P}} \newcommand {\R }[0]{\mathbb {R}} \newcommand {\Q }[0]{\mathbb {Q}} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\N }[0]{\mathbb N} \newcommand {\ph }[0]{\varphi } \newcommand {\vec }[0]{\mathbf } \newcommand {\d }[0]{\,d} \newcommand {\egmil }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_person.pdf}\hspace {0.5mm}}} \newcommand {\eghuntho }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_fish.pdf}\hspace {0.5mm}}} \newcommand {\egtentho }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_finger.pdf}\hspace {0.5mm}}} \newcommand {\egtho }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_lotus.pdf}\hspace {0.5mm}}} \newcommand {\eghun }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_scroll.pdf}\hspace {0.5mm}}} \newcommand {\egten }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_heel.pdf}\hspace {0.5mm}}} \newcommand {\egone }[1]{\multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_stroke.pdf}\hspace {0.5mm}}} \newcommand {\egyptify }[7]{ \multido {\i =1+1}{#1}{\includegraphics [scale=.1]{egyptian/egypt_person.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#2}{\includegraphics [scale=.1]{egyptian/egypt_fish.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#3}{\includegraphics [scale=.1]{egyptian/egypt_finger.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#4}{\includegraphics [scale=.1]{egyptian/egypt_lotus.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#5}{\includegraphics [scale=.1]{egyptian/egypt_scroll.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#6}{\includegraphics [scale=.1]{egyptian/egypt_heel.pdf}\hspace {0.5mm}} \multido {\i =1+1}{#7}{\includegraphics [scale=.1]{egyptian/egypt_stroke.pdf}\hspace {0.5mm}} \hspace {.5mm} } \newcommand {\timing }[1]{{\bf Suggested Timing: \hspace {2ex}} #1} \newcommand {\degree }[0]{\ensuremath {^\circ }} \newcommand {\celsius }[0]{\ifmmode ^\circ \mathrm {C}\else $^\circ $C\fi } \newcommand {\ohm }[0]{\ifmmode \Omega \else $\Omega $\fi } changed the theorem header of amsthm failed\MessageBreak }}}$

An art class is planning to make fences out of popsicle sticks to decorate different walls around the school. The art teacher would like to know how many popsicle sticks will be used for fences of various sizes. The class agrees that the length of the fence is the number of popsicle sticks used in the bottom row of sticks.

Why is the fence shown here considered to be of length 6? How many rods does it use?

Sketch a fence of length 8. Sketch a fence of length 1.

The art teacher would like to write a formula to find the number of popsicle sticks $S$ in a fence of length $L$ . In other words, we will use $L$ to represent the length of the fence and $S$ to represent the number of popsicle sticks.

Find four different formulas that could be used to relate $L$ and $S$ . In each case, explain how your formula is related to the diagram. Be sure to use the meanings of any operations to explain why they are part of your formula. For example, if your formula was $18L - 5L + 12$ you would use groups and objects to explain why you multiplied $18 \times L$ and $5 \times L$ , then you would use the meaning of subtraction to explain why you subtract $18L - 5L$ and then the meaning of addition to explain why you added $12$ to the result of $18L - 5L$ . (Note that this is not a correct formula for this problem, just an example so that you can see what to explain.)

Can we build such a fence using exactly $267$ popsicle sticks? Explain your thinking.

2025-11-30 02:26:12