2. $\delta$ and $\blacklozenge$ stand for numbers. they are related by a rule. what is the rule?
\begin{array}{|c|c|c|c|c|}
\hline
\delta & 2 & 3 & 5 & 8 \\
\hline
\blacklozenge & 7 & 16 & 34 & 61 \\
\hline
\end{array}
a $\blacklozenge = 6 \times \delta - 5$
b $\blacklozenge = 9 \times \delta - 11$
c $\blacklozenge = \delta \times \delta + 3$
d $\blacklozenge = 2 \times \delta \times \delta - 1$

3. jo wants to draw a right-angled triangle. she has placed two points on the grid. where could she place her third point? select all correct answers.
a (4,2) b (2,4) c (4,1) d (1,4)

4. dylan used this rule to work out the next number in a pattern: multiply the previous number by 8 and then add 3. the first 3 numbers in the pattern are 11, 91 and 731. what is the fifth number in the pattern?

use this information to answer questions 5 and 6. ellie made a pattern with pentagons using pins. she drew up a table to show the number of pins needed for different numbers of pentagons.
\begin{array}{|c|c|c|c|}
\hline
\text{number of pentagons} & 1 & 2 & 3 \\
\hline
\text{number of pins} & 5 & 8 & 11 \\
\hline
\end{array}
5. how many pins are needed for 7 pentagons?
6. how many pentagons can be made with 60 pins?

7. what is the next number in this pattern? 1, 2, 5, 26, 677, ...

8. different numbers of squares in a pattern
\begin{array}{|c|c|c|c|c|}
\hline
\text{number of squares} & 1 & 2 & 3 & 4 \\
\hline
\text{number of matches} & & & 8 & \\
\hline
\end{array}
which could be the rule to give the number of matches? select all correct answers.
a double the number of squares and add 2.
b multiply the number of squares by itself and then subtract 1.
c add 5 to the number of squares.
d multiply the number of squares by 4 and subtract 5.

9. sergio wrote down the first seven numbers in a pattern: 1, 1, 2, 3, 5, 8, 13. what is the first number in this pattern that is greater than 100?

10. a rule to calculate the temperature in degrees celsius ($^{\circ}\text{c}$) when the temperature in degrees fahrenheit ($^{\circ}\text{f}$) is known is: first subtract 32, then divide by 1.8. an old recipe says to set the oven to 410 $^{\circ}\text{f}$. what is this temperature in $^{\circ}\text{c}$?

11. the table shows the total number of bricks that can be stacked in different numbers of rows when following a particular pattern.
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{number of rows} & 1 & 2 & 3 & 4 & 5 \\
\hline
\text{number of bricks} & 1 & 3 & 6 & 10 & 15 \\
\hline
\end{array}
what is the minimum number of rows needed to stack a total of 66 bricks?

12. the first two numbers in a pattern are 8 and 55. ben knows that the rule for the pattern is to square the number and then either add or subtract (he cant remember which) a certain number. what is the third number in the pattern?

Question

2. $\delta$ and $\blacklozenge$ stand for numbers. they are related by a rule. what is the rule?
\begin{array}{|c|c|c|c|c|}
\hline
\delta & 2 & 3 & 5 & 8 \\
\hline
\blacklozenge & 7 & 16 & 34 & 61 \\
\hline
\end{array}
a $\blacklozenge = 6 \times \delta - 5$
b $\blacklozenge = 9 \times \delta - 11$
c $\blacklozenge = \delta \times \delta + 3$
d $\blacklozenge = 2 \times \delta \times \delta - 1$

3. jo wants to draw a right-angled triangle. she has placed two points on the grid. where could she place her third point? select all correct answers.
a (4,2) b (2,4) c (4,1) d (1,4)

4. dylan used this rule to work out the next number in a pattern: multiply the previous number by 8 and then add 3. the first 3 numbers in the pattern are 11, 91 and 731. what is the fifth number in the pattern?

use this information to answer questions 5 and 6. ellie made a pattern with pentagons using pins. she drew up a table to show the number of pins needed for different numbers of pentagons.
\begin{array}{|c|c|c|c|}
\hline
\text{number of pentagons} & 1 & 2 & 3 \\
\hline
\text{number of pins} & 5 & 8 & 11 \\
\hline
\end{array}
5. how many pins are needed for 7 pentagons?
6. how many pentagons can be made with 60 pins?

7. what is the next number in this pattern? 1, 2, 5, 26, 677, ...

8. different numbers of squares in a pattern
\begin{array}{|c|c|c|c|c|}
\hline
\text{number of squares} & 1 & 2 & 3 & 4 \\
\hline
\text{number of matches} & & & 8 & \\
\hline
\end{array}
which could be the rule to give the number of matches? select all correct answers.
a double the number of squares and add 2.
b multiply the number of squares by itself and then subtract 1.
c add 5 to the number of squares.
d multiply the number of squares by 4 and subtract 5.

9. sergio wrote down the first seven numbers in a pattern: 1, 1, 2, 3, 5, 8, 13. what is the first number in this pattern that is greater than 100?

10. a rule to calculate the temperature in degrees celsius ($^{\circ}\text{c}$) when the temperature in degrees fahrenheit ($^{\circ}\text{f}$) is known is: first subtract 32, then divide by 1.8. an old recipe says to set the oven to 410 $^{\circ}\text{f}$. what is this temperature in $^{\circ}\text{c}$?

11. the table shows the total number of bricks that can be stacked in different numbers of rows when following a particular pattern.
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{number of rows} & 1 & 2 & 3 & 4 & 5 \\
\hline
\text{number of bricks} & 1 & 3 & 6 & 10 & 15 \\
\hline
\end{array}
what is the minimum number of rows needed to stack a total of 66 bricks?

12. the first two numbers in a pattern are 8 and 55. ben knows that the rule for the pattern is to square the number and then either add or subtract (he cant remember which) a certain number. what is the third number in the pattern?

Accepted Answer

{ "quality": "clear", "question_count": 12, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Pattern Rules", "Coordinate Geometry", "Fibonacci Sequence" ], "current_concepts": [ "Pattern Rules", "Coordinate Geometry", "Fibonacci Sequence", "Arithmetic Sequences", "Quadratic Sequences" ] } ## Analyze questions 2 through 4 - **Question 2**: Find the rule relating $\Delta$ and $\blacklozenge$. Given pairs $(\Delta, \blacklozenge)$: $(2, 7)$, $(3, 16)$, $(5, 34)$, $(8, 61)$. Test option D: $\blacklozenge = 2 \times \Delta \times \Delta - 1$. For $\Delta = 2$: $2(2)^2 - 1 = 7$. For $\Delta = 3$: $2(3)^2 - 1 = 17 eq 16$. Test option C: $\blacklozenge = \Delta \times \Delta + 3$. For $\Delta = 3$: $3^2 + 3 = 12 eq 16$. Test option B: $\blacklozenge = 9 \times \Delta - 11$. For $\Delta = 2$: $9(2) - 11 = 7$. For $\Delta = 3$: $9(3) - 11 = 16$. For $\Delta = 5$: $9(5) - 11 = 34$. For $\Delta = 8$: $9(8) - 11 = 61$. Thus, B is correct. - **Question 3**: Right-angled triangle. Two points are plotted at $(1, 2)$ and $(4, 4)$. To form a right-angled triangle, the third point must form a right angle with the segment connecting $(1, 2)$ and $(4, 4)$. If the right angle is at the third point $P(x, y)$ with horizontal and vertical sides: Option A: $(4, 2)$ forms a right triangle with vertices $(1, 2)$, $(4, 4)$, and $(4, 2)$ (right angle at $(4, 2)$). Option D: $(1, 4)$ forms a right triangle with vertices $(1, 2)$, $(4, 4)$, and $(1, 4)$ (right angle at $(1, 4)$). Thus, A and D are correct. - **Question 4**: Rule: Multiply by 8, then add 3. $a_1 = 11$ $a_2 = 11 \times 8 + 3 = 91$ $a_3 = 91 \times 8 + 3 = 731$ $a_4 = 731 \times 8 + 3 = 5851$ $a_5 = 5851 \times 8 + 3 = 46811$ ## Analyze questions 5 through 8 - **Question 5**: Pattern of pentagons: $5, 8, 11, \dots$ Rule: $P(n) = 3n + 2$. For $n = 7$: $3(7) + 2 = 23$. - **Question 6**: Find $n$ when $P(n) = 60$. $3n + 2 = 60 \implies 3n = 58 \implies n = 19.33$. Since $n$ must be an integer, the maximum number of complete pentagons is 19. - **Question 7**: Pattern: $1, 2, 5, 26, 677, \dots$ Rule: $a_{n} = a_{n-1}^2 + 1$. Next term: $677^2 + 1 = 458329 + 1 = 458330$. - **Question 8**: Pattern of matches for squares. For $n = 3$, matches = 8. Test option B: Multiply by itself and subtract 1: $3^2 - 1 = 8$. Test option C: Add 5: $3 + 5 = 8$. Test option D: Multiply by 4 and subtract 5: $3 \times 4 - 5 = 7 eq 8$. Thus, B and C are correct. ## Analyze questions 9 through 12 - **Question 9**: Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$ First number greater than 100 is 144. - **Question 10**: Formula: $C = (F - 32) / 1.8$. For $F = 410$: $C = (410 - 32) / 1.8 = 378 / 1.8 = 210$. - **Question 11**: Triangular numbers: $1, 3, 6, 10, 15, \dots$ Formula: $S_n = \frac{n(n+1)}{2}$. We want $S_n = 66$: $\frac{n(n+1)}{2} = 66 \implies n(n+1) = 132 \implies n^2 + n - 132 = 0 \implies (n-11)(n+12) = 0 \implies n = 11$. - **Question 12**: First two numbers: 8 and 55. Rule: $a_n = a_{n-1}^2 \pm C$. Let's find $C$: $55 = 8^2 - C \implies 55 = 64 - C \implies C = 9$. So the rule is $a_n = a_{n-1}^2 - 9$. Third number: $55^2 - 9 = 3025 - 9 = 3016$. | No. | Answer | |---|---| | 2 | B | | 3 | A, D | | 4 | 46811 | | 5 | 23 | | 6 | 19 | | 7 | 458330 | | 8 | B, C | | 9 | 144 | | 10 | 210 | | 11 | 11 | | 12 | 3016 | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Pattern Rules" ] }

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