Analyze and Graph Number Patterns - 5th Grade Math

1. Generate the first $4$ terms: start $10$, rule "subtract $2$".

   $10, 8, 6, 4$. Start at $10$, repeatedly subtract $2$ to get each next term.

2. Form the ordered pair from term $4$ of: A start $0$ add $3$; B start $0$ add $6$.

   $(9, 18)$. Term $4$ of A is $9$, term $4$ of B is $18$.

3. In the patterns start $0$ add $3$ and start $0$ add $6$, what is the relationship between terms?

   Each term in pattern $2$ is $2$ times the corresponding term in pattern $1$. Pattern 2 adds $6$ while pattern 1 adds $3$, so ratio is $2:1$.

4. What does the rule "Multiply by $m$" mean for moving from one term to the next?

   Each term equals the previous term times $m$. Multiplication rule means multiplying by the same factor each time.

5. Generate the first three terms: start $1$, rule "Add $4$". What are terms $1$ to $3$?

   $1, 5, 9$. Start at $1$, then $1+4=5$, then $5+4=9$.

6. What is the relationship if Pattern B is always $5$ more than Pattern A?

   $B = A + 5$ for corresponding terms. Each B term equals its corresponding A term plus 5.

7. What are the first three ordered pairs $(A,B)$ for A "Start $2$, add $4$" and B "Start $1$, add $2$"?

   $(2,1),(6,3),(10,5)$. Pairs corresponding terms from A and B in order.

8. If pattern $2$ is always $5$ more than pattern $1$, what relationship describes the ordered pairs?

   $y=x+5$. Each $y$-value equals its $x$-value plus $5$.

9. What ordered pair should you form from term $n$ of Pattern A and term $n$ of Pattern B?

   $(A_n, B_n)$. Pairs the $n$th terms from each pattern as $(x, y)$ coordinates.

10. Identify the rule: ordered pairs $(1,4),(2,7),(3,10)$ follow what equation?

    $y=3x+1$. Each $y$ increases by $3$ as $x$ increases by $1$, with $y$-intercept $1$.

11. If the ordered pairs follow $y = x - 2$, what is the ordered pair when $x = 7$?

    $(7, 5)$. Substitute $x = 7$ into $y = x - 2$ to get $y = 7 - 2 = 5$.

12. Identify the missing term: start $2$, rule "add $5$"; sequence $2,7,12,\underline{\ \ },22$.

    $17$. Each term adds $5$ to the previous: $12+5=17$.

13. What ordered pair is formed from corresponding terms $x$ (pattern 1) and $y$ (pattern 2)?

    The ordered pair is $(x,y)$. First pattern value goes first, second pattern value goes second.

14. Pattern A: start $4$, add $2$. Pattern B: start $1$, add $2$. What is $B$ in terms of $A$?

    $B = A - 3$. A: $4,6,8,10$; B: $1,3,5,7$; B starts 3 lower than A.

15. Pattern A: start $1$, add $4$. Pattern B: start $1$, add $8$. What is $B$ in terms of $A$?

    $B = 2A - 1$. A: $1,5,9,13$; B: $1,9,17,25$; B grows twice as fast minus 1.

16. Which equation matches the ordered pairs from "Start $0$, add $3$" and "Start $0$, add $6$": $y=2x$ or $y=x+3$?

    $y=2x$. Each $y$-value is twice its $x$-value: $0=2×0$, $6=2×3$, $12=2×6$.

17. Identify the missing term: start $0$, rule "add $3$"; sequence $0,3,6,\underline{\ \ },12$.

    $9$. Each term adds $3$ to the previous: $6+3=9$.

18. What is the relationship between patterns A "Start $2$, add $4$" and B "Start $1$, add $2$" for corresponding terms?

    Each term in A is $2$ times the corresponding term in B. Check: $2=2×1$, $6=2×3$, $10=2×5$, $14=2×7$ confirms the pattern.

19. Which axis shows the first number in an ordered pair $(x,y)$ on a coordinate plane?

    The $x$-axis. The horizontal axis represents the first coordinate.

20. What is the first term if the starting number is $7$?

    The first term is $7$. The starting number becomes the first value in the sequence.

21. Identify the first five terms for the rule "Start $0$, add $6$".

    $0,6,12,18,24$. Starting at $0$, add $6$ repeatedly: $0+6=6$, $6+6=12$, etc.

22. Which ordered pair is correct for term $3$ if Pattern A term $3$ is $12$ and Pattern B term $3$ is $5$?

    $(12, 5)$. Pattern A value goes first (x), Pattern B value goes second (y).

23. What do the $x$-coordinate and $y$-coordinate represent when graphing two patterns?

    $x$ is the term from pattern 1; $y$ is the corresponding term from pattern 2. Standard convention: first pattern on $x$-axis, second on $y$-axis.

24. What is the relationship if Pattern B is always $3$ less than Pattern A?

    $B = A - 3$ for corresponding terms. Each B term equals its corresponding A term minus 3.

25. Which point matches "pattern $1$ term $4$ and pattern $2$ term $9$" on the coordinate plane?

    $(4,9)$. The ordered pair shows pattern 1's term first, pattern 2's term second.

26. What is an ordered pair made from two patterns, using corresponding terms?

    A pair $(x,y)$ where $x$ is from pattern 1 and $y$ is the matching term from pattern 2. First pattern gives $x$, second pattern gives $y$ at same position.

27. Choose the ordered pair that correctly matches term $4$ for patterns: A "Start $3$, add $5$" and B "Start $10$, add $5$".

    $(18,25)$. Term 4: A is $3+5×3=18$, B is $10+5×3=25$.

28. If pattern $2$ is always $3$ times pattern $1$, what relationship describes the ordered pairs?

    $y=3x$. Each $y$-value equals its $x$-value times $3$.

29. Identify the first four terms for pattern A: "Start $5$, add $2$".

    $5,7,9,11$. Start at $5$, add $2$ repeatedly: $5+2=7$, $7+2=9$, $9+2=11$.

30. What does the rule "Add $k$" mean for moving from one term to the next?

    Each term equals the previous term plus $k$. Addition rule means adding the same constant to generate each new term.

31. What is the relationship between corresponding terms if one pattern is "Start $0$, add $3$" and the other is "Start $0$, add $6$"?

    Each term in the second pattern is $2$ times the corresponding term in the first. Since $6=2×3$, each "add $6$" term equals $2×$ the "add $3$" term.

32. Generate $3$ ordered pairs: start $1$ add $2$ (pattern $1$) and start $3$ add $2$ (pattern $2$).

    $(1,3),(3,5),(5,7)$. Pattern 1: $1,3,5$; Pattern 2: $3,5,7$; pair corresponding positions.

33. Identify the first four terms for pattern A: "Start $2$, add $4$".

    $2,6,10,14$. Start at $2$, add $4$ repeatedly: $2+4=6$, $6+4=10$, $10+4=14$.

34. What are the first $4$ terms for start $10$, rule "subtract $2$"?

    $10,8,6,4$. Start at $10$, then subtract $2$ repeatedly: $10-2=8$, $8-2=6$, $6-2=4$.

35. If the ordered pairs follow $y = 3x$, what is $y$ when $x = 4$?

    $12$. Substitute $x = 4$ into $y = 3x$ to get $y = 3(4) = 12$.

36. Identify the relationship: A start $0$ add $3$; B start $0$ add $6$. How is $B_n$ related to $A_n$?

    $B_n = 2A_n$. Each B term is double its corresponding A term since $6 = 2 \times 3$.

37. What are the first $4$ terms for start $1$, rule "multiply by $2$"?

    $1,2,4,8$. Start at $1$, then double repeatedly: $1×2=2$, $2×2=4$, $4×2=8$.

38. What does the term "corresponding terms" mean when comparing two patterns?

    Terms in the same position (same step number) in each pattern. Terms at the same step number match up between patterns.

39. From patterns A: $1,5,9$ and B: $2,10,18$, what is the ordered pair for term $2$?

    $(5, 10)$. Term $2$ of A is $5$, term $2$ of B is $10$.

40. What are the first $4$ terms for start $0$, rule "multiply by $2$"?

    $0,0,0,0$. Multiplying $0$ by any number always gives $0$.

41. Identify the relationship if Pattern B is always $2$ times Pattern A for each term.

    $B = 2A$ for corresponding terms. Each B term equals its corresponding A term multiplied by 2.

42. Identify the missing value: Pattern A "Start $0$, add $4$" gives $0,4,8,12$; if $B=3A$, what is $B$ at term $3$?

    $24$. Term 3 of A is $8$, so $B=3×8=24$.

43. What is the meaning of a numerical pattern generated by a rule and a starting number?

    A sequence made by repeatedly applying a rule to a starting value. Each application of the rule creates the next term in order.

44. What are the first three ordered pairs for patterns "Start $0$, add $3$" and "Start $0$, add $6$"?

    $(0,0),(3,6),(6,12)$. Pairs corresponding terms: $(0,0)$, $(3,6)$, $(6,12)$.

45. Identify the relationship for A: start $1$ add $4$ and B: start $2$ add $8$ in terms of $A_n$.

    $B_n = 2A_n$. Each B term doubles its A term: $2 = 2(1)$, $10 = 2(5)$, $18 = 2(9)$.

46. Generate the first three terms: start $2$, rule "Add $8$". What are terms $1$ to $3$?

    $2, 10, 18$. Start at $2$, then $2+8=10$, then $10+8=18$.

47. What is a "corresponding term" when comparing two patterns with the same term number?

    Terms in the same position (same term number) in both sequences. Corresponding terms share the same position number in their sequences.

48. What is the relationship between corresponding terms if both patterns increase by $5$ each step, but starts are $3$ and $10$?

    Pattern B is always $7$ greater than pattern A. Same rate of change, but B starts $7$ higher: $10-3=7$.

49. Generate the first $4$ terms: start $2$, rule "add $5$".

    $2, 7, 12, 17$. Start at $2$, repeatedly add $5$ to get each next term.

50. On a coordinate plane, which axis shows the first pattern when graphing ordered pairs $(x,y)$ from two patterns?

    The $x$-axis shows the first pattern. Convention: $(x,y)$ means first pattern on $x$, second on $y$.

51. What is the starting number in a pattern, and where does it appear in the sequence?

    The first term; it is term $1$ of the sequence. The starting number becomes the first element in the pattern.

52. Pattern A: start $0$, add $2$. Pattern B: start $1$, add $4$. What is $B$ in terms of $A$?

    $B = 2A + 1$. A: $0,2,4,6$; B: $1,5,9,13$; B grows twice as fast plus 1.

53. What does the term "corresponding terms" mean for two patterns listed by term number?

    Terms in the same position (same term number) in each pattern. Terms at the same position number in both sequences match up.

54. If ordered pairs follow $y=x-4$, what is $y$ when $x=9$?

    $5$. Substitute $x=9$ into $y=x-4$: $y=9-4=5$.

55. Generate term $4$: start $0$, rule "Add $3$". What is the term $4$ value?

    $9$. Pattern: $0, 3, 6, 9$; term $4$ is $0 + 3(3) = 9$.

56. Which axis shows the second pattern values when graphing ordered pairs $(A,B)$?

    The $y$-axis shows Pattern B values. In ordered pairs $(A,B)$, B values go on the vertical axis.

57. Pattern A: start $0$, add $3$. Pattern B: start $0$, add $6$. What are the first $4$ ordered pairs $(A,B)$?

    $(0,0), (3,6), (6,12), (9,18)$. Pair corresponding terms: $(0,0)$, $(3,6)$, $(6,12)$, $(9,18)$.

58. Find the first $5$ terms for start $0$, rule "add $6$".

    $0, 6, 12, 18, 24$. Start at $0$, repeatedly add $6$ to get each next term.

59. Which equation matches the relationship if $B-A=-4$ for all corresponding terms?

    $B=A-4$. Rearranging $B-A=-4$ gives $B=A-4$.

60. Identify the next term: start $0$, rule "Add $3$", current term $9$.

    $12$. Apply the rule: $9 + 3 = 12$.

61. Identify the next term: start $1$, rule "Multiply by $2$", current term $8$.

    $16$. Apply the rule: $8 \times 2 = 16$.

62. Identify the first four terms for pattern B: "Start $1$, add $4$".

    $1,5,9,13$. Start at $1$, add $4$ repeatedly: $1+4=5$, $5+4=9$, $9+4=13$.

63. What point is formed by the $4^{\text{th}}$ term of Pattern A and the $4^{\text{th}}$ term of Pattern B?

    The ordered pair $(A_4, B_4)$. The subscript 4 indicates both terms are from position 4.

64. What is the meaning of "corresponding terms" in two patterns?

    Terms in the same position, such as $1$st with $1$st, $2$nd with $2$nd. Pairs terms by position in each sequence.

65. What is an ordered pair made from two patterns if $x$ is Pattern A and $y$ is Pattern B?

    A point $(x, y)$ using matching (corresponding) terms. First pattern gives $x$-coordinate, second gives $y$-coordinate.

66. Identify the next term: start $2$, rule "Add $5$", current term $17$.

    $22$. Apply the rule: $17 + 5 = 22$.

67. What are the first $4$ terms for start $1$, rule "add $4$"?

    $1,5,9,13$. Start at $1$, then add $4$ repeatedly: $1+4=5$, $5+4=9$, $9+4=13$.

68. On a coordinate plane, where do you plot the ordered pair $(3, 7)$: which axis gets each number?

    Plot $x = 3$ on the horizontal axis and $y = 7$ on the vertical axis. First coordinate is horizontal position, second is vertical position.

69. Identify the first five terms for the rule "Start $0$, add $3$".

    $0,3,6,9,12$. Starting at $0$, add $3$ repeatedly: $0+3=3$, $3+3=6$, etc.

70. Generate $3$ ordered pairs: start $0$ add $3$ (pattern $1$) and start $0$ add $6$ (pattern $2$).

    $(0,0),(3,6),(6,12)$. Pattern 1: $0,3,6$; Pattern 2: $0,6,12$; pair corresponding positions.

71. Which coordinate represents Pattern A and which represents Pattern B in $(A_n, B_n)$?

    $x$-coordinate is Pattern A; $y$-coordinate is Pattern B. Standard convention places first pattern on $x$-axis, second on $y$-axis.

72. Find the first $5$ terms for start $0$, rule "add $3$".

    $0, 3, 6, 9, 12$. Start at $0$, repeatedly add $3$ to get each next term.

73. What do the coordinates $(x,y)$ represent when graphing two patterns?

    $x$ is the pattern $1$ term and $y$ is the matching pattern $2$ term. First coordinate from first pattern, second from second pattern.

74. Generate term $5$: start $0$, rule "Add $6$". What is the term $5$ value?

    $24$. Pattern: $0, 6, 12, 18, 24$; term $5$ is $0 + 6(4) = 24$.

75. Pattern A: start $2$, add $3$. Pattern B: start $5$, add $3$. What is $B$ in terms of $A$?

    $B = A + 3$. A: $2,5,8,11$; B: $5,8,11,14$; B starts 3 higher than A.

76. Which axis shows the first pattern values when graphing ordered pairs $(A,B)$?

    The $x$-axis shows Pattern A values. In ordered pairs $(A,B)$, A values go on the horizontal axis.

77. If ordered pairs follow $y=2x$, what is $y$ when $x=7$?

    $14$. Substitute $x=7$ into $y=2x$: $y=2(7)=14$.

78. Identify the first four terms for pattern B: "Start $1$, add $2$".

    $1,3,5,7$. Start at $1$, add $2$ repeatedly: $1+2=3$, $3+2=5$, $5+2=7$.

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