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v1 Algebra, STD2 A2 2012 HSC 8 MC

Dots were used to create a pattern. The first three shapes in the pattern are shown. 
 

 The number of dots used in each shape is recorded in the table. 

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape $(S)$} \rule[-1ex]{0pt}{0pt} &\;\;\;  1 \;\;\; & \;\; \;2  \;\;\; &   \;\;\; 3 \;\;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of dots $(N)$} \rule[-1ex]{0pt}{0pt} &\;\;\;  8 \;\;\; & \;\; \;10  \;\;\; &   \;\; \;12\; \;\; \\
\hline
\end{array}

How many dots would be required for Shape 182?

  1. 363
  2. 370
  3. 546
  4. 1092
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\(B\)

Show Worked Solution

\(\text{Linear relationship where}\)

\(N=6+(2\times S)\)

\(\text{When}\ \ S=182\)

\(N\) \(=6+(2\times 182)\)
  \(=370\)

  
\(\Rightarrow B\)

v1 Algebra, STD2 A2 2007 HSC 18 MC

Art started to make this pattern of shapes using matchsticks.
  

 

If the pattern of shapes is continued, which shape would use exactly 416 matchsticks?

  1. Shape 83
  2. Shape 103
  3. Shape 104
  4. Shape 138
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\(D\)

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape}\ \textit(S) \rule[-1ex]{0pt}{0pt}\ \ &\  \ 1\ \ &\ \ 2\ \ &\ \ 3\ \  \\
\hline
\rule{0pt}{2.5ex} \text{Matches}\ \textit(M) \rule[-1ex]{0pt}{0pt} \ \ & \ \ 5\ \ &\ \ 8\ \ &\ \ 11\ \  \\
\hline
\end{array}

\(\text{Equation rule:}\)

\(M=3S+2\)

\(\text{Find}\ \ S\ \text{when}\ \ M=416:\)

\(416\) \(=3S+2\)
\(3S\) \(=414\)
\(S\) \(=138\)

 
\(\therefore\ \text{The 138th shape uses 416 matchsticks.}\)

\(\Rightarrow D\)

v1 Algebra, STD2 A2 2011 HSC 23b

Sticks were used to create the following pattern. 
  

The number of sticks used is recorded in the table.

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape $(S)$} \rule[-1ex]{0pt}{0pt} & \;\;\; 1 \;\;\; & \;\;\; 2 \;\;\; & \;\;\; 3 \;\;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of sticks $(N)$}\; \rule[-1ex]{0pt}{0pt} & \;\;\; 6 \;\;\; & \;\;\; 10 \;\;\; & \;\;\; 14 \;\;\; \\
\hline
\end{array}

  1. Draw Shape 4 of this pattern.  (1 mark)

  2. How many sticks would be required for Shape 128?    (1 mark)

  3. Is it possible to create a shape in this pattern using exactly 609 sticks?

     

    Show suitable calculations to support your answer.    (2 marks)

Show Answers Only
  1. \(\text{See Worked Solutions.}\)
  2. \(514\)
  3. \(\text{No (See worked solution)}\)
Show Worked Solution

  i.    \(\text{Shape 4 is shown below:}\)

ii.    \(\text{Since}\ \ N=2+4S\)

♦ Mean mark 48%.
MARKER’S COMMENT: Students should attempt to find a “rule” in such questions, and use this formula to solve the question, as per the Worked Solution.  
\(\text{If }S\) \(=128\)
\(N\) \(=2+(4\times 128)\)
  \(=514\)

 

iii.    \(609\) \(=2+4S\)
  \(4S\) \(=607\)
  \(S\) \(=151.75\)

    
\(\text{Since}\ S\ \text{is not a whole number, 609 sticks}\)

\(\text{will not create a shape in this pattern.}\)

v1 Algebra, STD2 A2 2017 HSC 20 MC

A pentagon is created using matches.

By adding more matches, a row of two pentagons is formed.

Continuing to add matches, a row of three pentagons can be formed.

Continuing this pattern, what is the maximum number of complete pentagons that can be formed if 230 matches in total are available?

  1. 55
  2. 56
  3. 57
  4. 58
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\(C\)

Show Worked Solution

\(\text{1 pentagon:}\ 5+4\times 0=5\)

\(\text{2 pentagons:}\ 5+4\times 1=9\)

\(\text{3 pentagons:}\ 5+4\times 2 = 13\)

\(\vdots\)

\(n\ \text{pentagons:}\ 5 + 4(n – 1)\)

\(5+4(n – 1)\) \(=230\)
\(4n-4\) \(=225\)
\(4n\) \(=229\)
\(n\) \(=57.25\)

 

\(\text{Complete pentagons possible}\ =57\)

\(\Rightarrow C\)

v1 Algebra, STD2 A2 SM-Bank 20

Brett uses matchsticks to make a pattern of shapes, as shown in the table below.
  

   How many sticks (\(S\)) will be needed to make Shape Number 24?  (2 marks)

Show Answers Only

\(121\)

Show Worked Solution

\(\text{Shape 1:}\ \ S=5\times 1+1=6\)

\(\text{Shape 2:}\ \ S=5\times 2+1=11\)

\(\vdots\)

\(\text{Shape}\ N:\ \ S=5N+1\)
 

\(\therefore\ \text{Sticks required when}\ \ N=24\)

\(=24\times 5+1\)

\(=121\)