smc-4216-15-Patterns

Linear Relationships, SM-Bank 039 MC

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

The equation used to show the relationship between $T$ and $N$ is

$T = N + 1$
$T = 6N$
$T = 6N+2$
$T = 6N-4$

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$D$

Show Worked Solution

$T\ \text{increases by 6 each shape.}$

$\text{Consider}\ T = 6N – 4:$

$\text{When}\ \ N = 1,\ T = 6\times 1 − 4 = 2$

$\text{When}\ \ N = 2, \ T = 6\times − 4 = 8$

$\text{When}\ \ N = 3,\ T = 6\times − 4 = 14$

$\therefore T = 6N − 4\ \text{is correct}$

$\Rightarrow D$

Linear Relationships, SM-Bank 032 MC

Which rule correctly describes the pattern below?

$\text{The number of pins}=2\times \text{The number of squares}+3$
$\text{The number of pins}=3\times \text{The number of squares}+1$
$\text{The number of pins}=1\times \text{The number of squares}+3$
$\text{The number of pins}=4\times \text{The number of squares}$

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$B$

Show Worked Solution

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Number of squares} \rule[-1ex]{0pt}{0pt} & 1 & 2 & 3 \\
\hline
\rule{0pt}{2.5ex} \text{Number of pins} \rule[-1ex]{0pt}{0pt} & 4 & 7 & 10 \\
\hline
\end{array}

$\therefore\ \text{The number of pins}=3\times \text{Number of squares}+1$

$\Rightarrow B$

Linear Relationships, SM-Bank 031 MC

Which rule correctly describes the pattern below?

$\text{The number of pins}=\text{The number of triangles}+3$
$\text{The number of pins}=\text{The number of triangles}+5$
$\text{The number of pins}=\text{The number of triangles}\times 2$
$\text{The number of pins}=\text{The number of triangles}\times 3$

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$D$

Show Worked Solution

$\text{Number of triangles }(t)$	$\ \ 1\ \ $	$\ \ 2\ \ $	$\ \ 3\ \ $	$\ \ ….\ \ $
$\text{Number of pins }(p)$	$\ \ 3\ \ $	$\ \ 6\ \ $	$\ \ 9\ \ $	$\ \ ….\ \ $

$\therefore\ \text{The number of pins}=\text{Number of triangles}\times 3$

$\Rightarrow D$

Linear Relationships, SM-Bank 030

Michael is making a geometric pattern using sticks to make pentagons.

The first 3 shapes in the pattern are shown below.

Draw the next shape in the pattern. (1 mark)

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Complete the table of values using the pattern. (2 marks)

Number of pentagons $(\large p)$ $\ \ 1\ \ $ $\ \ 2\ \ $ $\ \ 3\ \ $ $\ \ 4\ \ $

Number of sticks $(\large s)$ $\ \ 5\ \ $
Write the rule connecting the number of sticks $(s)$ to the number of pentagons $(p)$. (2 marks)

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How many sticks will be needed to make $12$ pentagons? (2 marks)

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a. $\text{Shape number }4$

b. $\text{Table of values}$

$\text{Number of pentagons }(\large p)$	$\ \ 1\ \ $	$\ \ 2\ \ $	$\ \ 3\ \ $	$\ \ 4\ \ $
$\text{Number of sticks }(\large s)$	$\ \ 5\ \ $	$\ \ 9\ \ $	$\ \ 13\ \ $	$\ \ 17\ \ $

c. $s=4\times p+1$

d. $49$

Show Worked Solution

a. $\text{Shape number }4$

b. $\text{Table of values}$

$\text{Number of pentagons }(\large p)$	$\ \ 1\ \ $	$\ \ 2\ \ $	$\ \ 3\ \ $	$\ \ 4\ \ $
$\text{Number of sticks }(\large s)$	$\ \ 5\ \ $	$\ \ 9\ \ $	$\ \ 13\ \ $	$\ \ 17\ \ $

c. $\text{Rule: The number of sticks}=4\times \text{(the number of pentagons)}+1$

$\therefore\ \text{Rule: }\ s=4\times p+1$

d. $\text{Find the value of }s\ \text{when }p=12$

$s=4\times p+1=4\times 12+1=49$

Linear Relationships, SM-Bank 029

Michael is making a geometric pattern using pins to form triangles.

The first 3 shapes in the pattern are shown below.

Draw the next shape in the pattern. (1 mark)

--- 6 WORK AREA LINES (style=lined) ---
Complete the table of values using the pattern. (2 marks)

Number of triangles $(t)$ $\ \ 1\ \ $ $\ \ 2\ \ $ $\ \ 3\ \ $ $\ \ 4\ \ $

Number of pins $(p)$ $\ \ 3\ \ $
Write the rule connecting the number of pins $(p)$ to the number of triangles $(t)$. (2 marks)

--- 2 WORK AREA LINES (style=lined) ---
How many pins will be needed to make $25$ triangles? (2 marks)

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a. $\text{Shape number }4$

b. $\text{Table of values}$

$\text{Number of triangles }(t)$	$\ \ 1\ \ $	$\ \ 2\ \ $	$\ \ 3\ \ $	$\ \ 4\ \ $
$\text{Number of pins }(p)$	$\ \ 3\ \ $	$\ \ 5\ \ $	$\ \ 7\ \ $	$\ \ 9\ \ $

c. $p=2\times t+1$

d. $51$

Show Worked Solution

a. $\text{Shape number }4$

b. $\text{Table of values}$

$\text{Number of triangles }(t)$	$\ \ 1\ \ $	$\ \ 2\ \ $	$\ \ 3\ \ $	$\ \ 4\ \ $
$\text{Number of pins }(p)$	$\ \ 3\ \ $	$\ \ 5\ \ $	$\ \ 7\ \ $	$\ \ 9\ \ $

c. $\text{Rule: The number of pins}=2\times \text{(the number of triangles)}+1$

$\therefore\ \text{Rule: }\ p=2\times t+1$

d. $\text{Find the value of }p\ \text{when }t=25$

$p=2\times t+1=2\times 25+1=51$

Linear Relationships, SM-Bank 024

The table below has a pattern. The top and bottom numbers are connected by a rule.

Top Number	$1$	$2$	$3$	$4$
Bottom Number	$0$	$-1$	$-2$	$-3$

What is the rule connecting the top number and the bottom number? (2 marks)

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What is the bottom number when the top number is $21$? (2 marks)

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a. $\text{Bottom number}=\text{Top number}\ ÷ \ 3$

b. $-2$

Show Worked Solution

Top Number	$1$	$2$	$3$	$4$
Bottom Number	$1-1=0$	$1-2=-1$	$1-3=-2$	$1-4=-3$

$\text{Rule: Bottom number}=1-\text{Top number}$

b. $\text{Bottom number}=1-\text{Top number}=1-21=-20$

Linear Relationships, SM-Bank 023

The table below has a pattern. The top and bottom numbers are connected by a rule.

Top Number	21	18	15	12
Bottom Number	7	6	5	4

What is the rule connecting the top number and the bottom number? (2 marks)

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What is the bottom number when the top number is $-6$? (2 marks)

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a. $\text{Bottom number}=\text{Top number}\ ÷ \ 3$

b. $-2$

Show Worked Solution

Top Number	$21$	$18$	$15$	$12$
Bottom Number	$21\ ÷\ 3=7$	$18\ ÷\ 3=6$	$15\ ÷\ 3=5$	$12\ ÷\ 3=4$

$\text{Rule: Bottom number}=\text{Top number}\ ÷\ 3$

b. $\text{Bottom number}=\text{Top number}\ ÷\ 3=-6\ ÷\ 3=-2$

Linear Relationships, SM-Bank 022

The table below has a pattern. The top and bottom numbers are connected by a rule.

Top Number	2	4	6	8
Bottom Number	8	16	24	32

What is the rule connecting the top number and the bottom number? (2 marks)

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What is the bottom number when the top number is 15? (2 marks)

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a. $\text{Bottom number}=4\times \text{Top number}$

b. $60$

Show Worked Solution

Top Number	$2$	$4$	$6$	$8$
Bottom Number	$4\times 2=8$	$4\times 4=16$	$4\times 6=24$	$4\times 8=32$

$\text{Rule: Bottom number}=4\times \text{Top number}$

b. $\text{Rule: Bottom number}=4\times \text{Top number}=4\times 15=60$

Linear Relationships, SM-Bank 022

Sabre is saving to buy a new skateboard.

After one week she has saved $11.

She then saves the same amount of money each week.

Week	1	2	3	4
Total Amount Saved	$11	$18	$25	$32

State the rule linking the week and the total amount saved. (2 marks)

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How much money will Sabre have saved by the end of week 10? (2 marks)

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a. $\text{Rule: Amount saved}=$4 + \text{week}\times 7$

b. $$74$

Show Worked Solution

a. $\text{Total at end of week }1= $11$

$\therefore\ \text{After week 1 savings increase by }$7\ \text{per week}$

$\therefore\ \text{Total at end of week 2}=$4 + 2\times 7= $18$

$\therefore\ \text{Total at end of week 3}=$4 + 3\times 7= $25$

$\therefore\ \text{Total at end of week 4}=$4 + 4\times 7= $32$

$\therefore\ \text{Rule: Amount saved}=$4 + \text{week}\times 7$

b. $\text{Total savings at end of week}\ 10$

$= 4 + 10\times 7$

$= $74$

Linear Relationships, SM-Bank 020 MC

Jerry's wage is calculated using an amount per hour plus a travel allowance.

This table shows some of Jerry's wage amounts.

Hours	1	2	3	4
Wage	$85	$140	$195	$250

How are Jerry's wages calculated?

$40 per hour + $35 travel allowance
$60 per hour + $25 travel allowance
$55 per hour + $30 travel allowance
$45 per hour + $40 travel allowance

Show Answers Only

$C$

Show Worked Solution

$\text{Testing Option C equation with the table values:}$

$85$	$=1\times 55+30\ \ \checkmark$
$140$	$=2\times 55+30\ \ \checkmark$
$195$	$=3\times 55+30\ \ \checkmark$
$250$	$=4\times 55+30\ \ \checkmark$

$\Rightarrow C$

Linear Relationships, SM-Bank 019 MC

A plumber calculates the price of a job using a service fee and an amount per hour.

This table shows some of the job prices.

Hours	1	2	3	4
Job price	$90	$130	$170	$210

How are the jobs calculated?

$50 service fee + $40 per hour
$58 service fee + $32 per hour
$60 service fee + $30 per hour
$70 service fee + $20 per hour

Show Answers Only

$A$

Show Worked Solution

$\text{Testing Option A equation with the table values:}$

$90$	$= 50 + 1\times 40\ \ \checkmark$
$130$	$= 50 + 2\times 40\ \ \checkmark$
$170$	$= 50 + 3\times 40\ \ \checkmark$
$210$	$= 50 + 4\times 40\ \ \checkmark$

$\Rightarrow A$

Linear Relationships, SM-Bank 018 MC

Jennifer had 20 cupcakes for sale at the beginning of the day. The table shows the number of cupcakes at the beginning of each hour.

Hour	0	1	2	3
Cupcakes	20	16	12	8

The table also shows a pattern in the number of cupcakes sold. The correct pattern connecting the hour and the number of cupcakes is:

$20-\text{Hour}\times 1$
$20-\text{Hour}\times 4$
$20+\text{Hour}\times 2$
$19+\text{Hour}$

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$B$

Show Worked Solution

$\text{Consider Option B }:\ 20-\text{Hour}\times 4$

$\text{Hour 0}\longrightarrow$	$20-4\times 0=20$	$\checkmark$
$\text{Hour 1}\longrightarrow$	$20-4\times 1=16$	$\checkmark$
$\text{Hour 2}\longrightarrow$	$20-4\times 2=12$	$\checkmark$
$\text{Hour 3}\longrightarrow$	$20-4\times 3=8$	$\checkmark$

$\Rightarrow B$

Linear Relationships, SM-Bank 017 MC

This table shows the growth of a plant, in centimetres, over a 4 week period.

Week	1	2	3	4
Growth (cm)	3	4	5	6

The table also shows a pattern in the growth of the plant. The correct pattern connecting the week and the growth is:

$\text{Week}\times 3$
$\text{Week}\times 4-1$
$\text{Week}\times 2+1$
$\text{Week}+2$

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$D$

Show Worked Solution

$\text{Consider Option D }:\ \text{Week}+2$

$\text{Week 1}\longrightarrow$	$1+2=3$	$\checkmark$
$\text{Week 2}\longrightarrow$	$2+2=4$	$\checkmark$
$\text{Week 3}\longrightarrow$	$3+2=5$	$\checkmark$
$\text{Week 4}\longrightarrow$	$4+2=6$	$\checkmark$

$\Rightarrow D$

Linear Relationships, SM-Bank 016 MC

This chart shows the longest run, in kilometres, that Deek ran each week over 4 weeks.

Week	1	2	3	4
Longest Run (km)	8	11	14	17

The chart also shows a pattern in Deek's running. The correct pattern connecting the week and the longest run is:

$\text{Week}\times 8$
$\text{Week}\times 2+6$
$\text{Week}\times 3+5$
$\text{Week}+7$

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$C$

Show Worked Solution

$\text{Consider Option C }:\ \text{Week}\times 3+5$

$\text{Week 1}\longrightarrow$	$1\times 3+5=8$	$\checkmark$
$\text{Week 2}\longrightarrow$	$2\times 3+5=11$	$\checkmark$
$\text{Week 3}\longrightarrow$	$3\times 3+5=14$	$\checkmark$
$\text{Week 4}\longrightarrow$	$4\times 3+5=17$	$\checkmark$

$\Rightarrow C$

Linear Relationships, SM-Bank 015 MC

The table below has a pattern. The top and bottom numbers are connected by a rule.

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Top number} \rule[-1ex]{0pt}{0pt} &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ &\ \ 4\ \ & \ldots &\ \ ?\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Bottom number} \rule[-1ex]{0pt}{0pt} & 3 & 6 & 9 & 12 & \ldots & 27 \\
\hline
\end{array}

What is the top number when the bottom number is 27?

$5$
$6$
$9$
$19$

Show Answers Only

$C$

Show Worked Solution

$\text{Top number}\times 3 =\ \text{Bottom number}$

$\text{Top number}\times 3 = 27$

$\therefore\ \text{Top number}\ = \dfrac{27}{3}=9$

$\Rightarrow C$

Linear Relationships, SM-Bank 014 MC

The table below has a pattern. The top and bottom numbers are connected by a rule.

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Top Number} \rule[-1ex]{0pt}{0pt} & 1 & 2 & 3 & 4 & ... & ? \\
\hline
\rule{0pt}{2.5ex} \text{Bottom Number} \rule[-1ex]{0pt}{0pt} & 4 & 8 & 12 & 16 & ... & 28 \\
\hline
\end{array}

What is the top number when the bottom number is 28?

$5$
$7$
$12$
$17$

Show Answers Only

$B$

Show Worked Solution

$\text{Top number}\ \times 4 =\ \text{Bottom number}$

$\text{Top number}\ \times 4 = 28$

$\therefore\ \text{Top number}\ = \dfrac{28}{4}=7$

$\Rightarrow B$

\(\text{Number of triangles }(t)\)	\(\ \ 1\ \ \)	\(\ \ 2\ \ \)	\(\ \ 3\ \ \)	\(\ \ ….\ \ \)
\(\text{Number of pins }(p)\)	\(\ \ 3\ \ \)	\(\ \ 6\ \ \)	\(\ \ 9\ \ \)	\(\ \ ….\ \ \)

Number of pentagons \((\large p)\)	\(\ \ 1\ \ \)	\(\ \ 2\ \ \)	\(\ \ 3\ \ \)	\(\ \ 4\ \ \)
Number of sticks \((\large s)\)	\(\ \ 5\ \ \)

\(\text{Number of pentagons }(\large p)\)	\(\ \ 1\ \ \)	\(\ \ 2\ \ \)	\(\ \ 3\ \ \)	\(\ \ 4\ \ \)
\(\text{Number of sticks }(\large s)\)	\(\ \ 5\ \ \)	\(\ \ 9\ \ \)	\(\ \ 13\ \ \)	\(\ \ 17\ \ \)

\(\text{Number of pentagons }(\large p)\)	\(\ \ 1\ \ \)	\(\ \ 2\ \ \)	\(\ \ 3\ \ \)	\(\ \ 4\ \ \)
\(\text{Number of sticks }(\large s)\)	\(\ \ 5\ \ \)	\(\ \ 9\ \ \)	\(\ \ 13\ \ \)	\(\ \ 17\ \ \)

Number of triangles \((t)\)	\(\ \ 1\ \ \)	\(\ \ 2\ \ \)	\(\ \ 3\ \ \)	\(\ \ 4\ \ \)
Number of pins \((p)\)	\(\ \ 3\ \ \)

Top Number	\(1\)	\(2\)	\(3\)	\(4\)
Bottom Number	\(0\)	\(-1\)	\(-2\)	\(-3\)

Top Number	\(1\)	\(2\)	\(3\)	\(4\)
Bottom Number	\(1-1=0\)	\(1-2=-1\)	\(1-3=-2\)	\(1-4=-3\)

Top Number	\(21\)	\(18\)	\(15\)	\(12\)
Bottom Number	\(21\ ÷\ 3=7\)	\(18\ ÷\ 3=6\)	\(15\ ÷\ 3=5\)	\(12\ ÷\ 3=4\)

Top Number	\(2\)	\(4\)	\(6\)	\(8\)
Bottom Number	\(4\times 2=8\)	\(4\times 4=16\)	\(4\times 6=24\)	\(4\times 8=32\)