Show work:

Ans: _______

18. \(\frac{2m + 1}{5}\) \(m = 7\)

Show work:

Ans: _______

4 Level 4: Equations & Applied Formulas

Verify or solve for the unknown

19. Check Solution: Is \(t = 7\) a solution to \(3t - 5 = 16\)?

Show substitution on left side:

Circle one: [ YES / NO ]

20. Temperature Formula: \(F = \frac{9}{5}C + 32\). Find \(F\) when \(C = 20^\circ\text{C}\).

Show work:

\(F =\) _________ \(^\circ\text{F}\)

Quick Check Exit Ticket (Last 7 Minutes)

Teacher Score: [      / 3 ]

A. Evaluate: \(x^2 + 2x\)
when \(x = -3\)

Ans: ______

B. Evaluate: \(\frac{2k + 6}{4}\)
when \(k = 5\)

Ans: ______

C. Check: \(4y - 3 = 15\)
Is \(y = 4\) a solution?

[ YES / NO ]

Fractions Scaffold: Instruct students to replace the fraction bar with \(\div\) and brackets: \([\,numerator\,] \div denominator\).

Early Finishers: Challenge students to create an expression with variables \(x=-2\) and \(y=3\) that results in exactly zero, or evaluate \(\frac{x^2 - y^2}{x - y}\).

Substitution Mastery • 45-Minute Lesson Guide Standards: CCSS.MATH.CONTENT.6.EE.A.2.C • 7.EE.B.3 • 8.EE.C.7

Answer Key & Diagnostic Guide

Fractions, Formulas & Exit Ticket Solutions

Diagnostic Analysis for Common Fraction Stumbling Blocks

Page 2 of 2

3 Level 3: Fractions & Rational Expressions Solutions

Primary Lesson Target

13. \(\frac{x}{3} + 2\) at \(x = 9\)

\(= \frac{9}{3} + 2 = 3 + 2\)

\(= \mathbf{5}\)

Divide fraction term first.

14. \(\frac{n}{2} + 7\) at \(n = 10\)

\(= \frac{10}{2} + 7 = 5 + 7\)

\(= \mathbf{12}\)

Divide before adding constant.

15. \(\frac{a+b}{2}\) (\(a=8, b=12\))

\(= \frac{8 + 12}{2} = \frac{20}{2}\)

\(= \mathbf{10}\)

Add numerator before dividing!

16. \(\frac{24}{y} - 3\) at \(y = 4\)

\(= \frac{24}{4} - 3 = 6 - 3\)

\(= \mathbf{3}\)

Variable in denominator.

17. \(\frac{w}{5} + 8\) at \(w = -15\)

\(= \frac{-15}{5} + 8 = -3 + 8\)

\(= \mathbf{5}\)

\(-15 \div 5 = -3\), then add 8.

18. \(\frac{2m + 1}{5}\) at \(m = 7\)

\(= \frac{2(7) + 1}{5} = \frac{15}{5}\)

\(= \mathbf{3}\)

Simplify top completely: \(14+1=15\).

4 Level 4: Equations & Applied Formulas Solutions

19. Is \(t = 7\) a solution to \(3t - 5 = 16\)?

Left side: \(3(7) - 5 = 21 - 5 = 16\).
Right side: \(16\). Since \(16 = 16\), equation holds.

Answer: YES

20. \(F = \frac{9}{5}C + 32\) at \(C = 20^\circ\text{C}\)

\(F = \frac{9}{5}(20) + 32 = 9(4) + 32 = 36 + 32 = 68\)

Answer: \(F = \mathbf{68^\circ\text{F}}\)

Quick Check Exit Ticket Solutions (Grading Rubric)

1 pt each • Total 3 pts

Item A: \(x^2 + 2x\) at \(x = -3\)

\((-3)^2 + 2(-3)\)
\(= 9 - 6 = \mathbf{3}\)

Common error: \(-9-6 = -15\)

Item B: \(\frac{2k+6}{4}\) at \(k = 5\)

\(\frac{2(5) + 6}{4} = \frac{10+6}{4}\)
\(= \frac{16}{4} = \mathbf{4}\)

Common error: \(2.5 + 6 = 8.5\)

Item C: Check \(4y - 3 = 15\) (\(y = 4\))

\(4(4) - 3 = 16 - 3 = 13\)
Since \(13 \neq 15\), NO

Answer: NO

Diagnostic Action: If student misses Item B, assign mini-whiteboard practice grouping numerators: \(\frac{\text{sum}}{\text{denom}}\).

Mastery Benchmark: 3/3 = Proficient • 2/3 = Minor Review • 0-1/3 = Re-teach

Ans:

32.\(y^2 - 10\) when \(y = -5\)

Ans:

33.\(x^2 + 4x\) when \(x = -2\)

Ans:

34.\(x^2 - 3x\) when \(x = -4\)

Ans:

35.\(-x^2 + 10\) when \(x = 3\)

Ans:

36.\(a - b\) when \(a = 7, b = -3\)

Ans:

37.\(p - q\) when \(p = -4, q = 6\)

Ans:

38.\(m - n\) when \(m = -5, n = -8\)

Ans:

39.\(2x - 3y\) when \(x = -2, y = 4\)

Ans:

40.\(-5a + 2b\) when \(a = -3, b = -4\)

Ans:

41.\(x^2 + y^2\) when \(x = -3, y = -4\)

Ans:

42.\(a^2 - b^2\) when \(a = -5, b = 3\)

Ans:

43.\(3x^2 + 2x\) when \(x = -2\)

Ans:

44.\(-2k^2 + 5\) when \(k = -3\)

Ans:

45.\((x + 4)^2\) when \(x = -7\)

Ans:

46.\((y - 3)^2\) when \(y = -2\)

Ans:

47.\(4 - 3x^2\) when \(x = -2\)

Ans:

48.\(ab + c\) when \(a=-3, b=-4, c=-5\)

Ans:

49.\(x^3 + 10\) when \(x = -2\)

Ans:

50.\(2x^2 - 5x + 3\) when \(x = -1\)

Ans:

Part 2 of 4 • Halfway Point (50/100) Turn to Page 3: Fraction Intensive (51–75) ➔

Skill Sprint 100

Part 3: Fraction & Rational Intensive (Problems 51–75)

Page 3 of 4

Key Reminder: The horizontal fraction bar acts as a grouping symbol. Simplify top first!

No decimal panic!

51.\(\frac{x}{2} + 5\) when \(x = 8\)

Ans:

52.\(\frac{x}{3} - 4\) when \(x = 15\)

Ans:

53.\(\frac{n}{4} + 9\) when \(n = 20\)

Ans:

54.\(12 - \frac{y}{2}\) when \(y = 10\)

Ans:

55.\(\frac{18}{a} + 3\) when \(a = 3\)

Ans:

56.\(\frac{30}{b} - 2\) when \(b = 5\)

Ans:

57.\(\frac{x + 6}{2}\) when \(x = 8\)

Ans:

58.\(\frac{y - 4}{3}\) when \(y = 19\)

Ans:

59.\(\frac{2a + 1}{5}\) when \(a = 7\)

Ans:

60.\(\frac{3m - 2}{4}\) when \(m = 6\)

Ans:

61.\(\frac{a + b}{2}\) when \(a = 11, b = 9\)

Ans:

62.\(\frac{x - y}{4}\) when \(x = 23, y = 7\)

Ans:

63.\(\frac{m + n}{p}\) when \(m=14, n=10, p=3\)

Ans:

64.\(\frac{x}{4} + \frac{y}{3}\) when \(x = 8, y = 12\)

Ans:

65.\(\frac{2a}{3} - 1\) when \(a = 9\)

Ans:

66.\(\frac{x}{5} + 6\) when \(x = -15\)

Ans:

67.\(\frac{y}{4} - 5\) when \(y = -20\)

Ans:

68.\(\frac{x + 10}{2}\) when \(x = -4\)

Ans:

69.\(\frac{w - 6}{5}\) when \(w = -9\)

Ans:

70.\(\frac{-24}{k}\) when \(k = 6\)

Ans:

71.\(\frac{x^2}{3} + 2\) when \(x = 6\)

Ans:

72.\(\frac{a^2 + 1}{5}\) when \(a = -7\)

Ans:

73.\(\frac{3x + 2y}{2}\) when \(x = 4, y = 5\)

Ans:

74.\(\frac{10}{x} + \frac{12}{y}\) when \(x = 2, y = 3\)

Ans:

75.\(\frac{5(F - 32)}{9}\) when \(F = 68\)

Ans:

Part 3 of 4 • Fractions Conquered (75/100) Turn to Page 4: Formulas & Equation Checks (76–100) ➔

Skill Sprint 100

Part 4: Formulas, Geometry & Solutions (Problems 76–100)

Page 4 of 4

Geometry Formulas • Multi-Variable Systems • Equation True/False

Sprint to the finish!

76.\(2x + 3y\) when \(x = 5, y = 4\)

Ans:

77.\(4a - 5b\) when \(a = 6, b = 2\)

Ans:

78.\(xy + 7\) when \(x = 3, y = 8\)

Ans:

79.\(2xy - 9\) when \(x = 4, y = 5\)

Ans:

80.\(a^2 + b^2\) when \(a = 3, b = 4\)

Ans:

81.\(x^2 - y^2\) when \(x = 8, y = 6\)

Ans:

82.\(3(x + 2y)\) when \(x = 4, y = 3\)

Ans:

83.\(2(3a - b)\) when \(a = 5, b = 4\)

Ans:

84.\(x(y + z)\) when \(x=3, y=4, z=5\)

Ans:

85.\(abc\) when \(a=2, b=3, c=7\)

Ans:

86.\(d = rt\) when \(r = 55, t = 4\)

\(d =\)

87.\(A = lw\) when \(l = 15, w = 8\)

\(A =\)

88.\(P = 2l + 2w\) when \(l = 14, w = 9\)

\(P =\)

89.\(V = lwh\) when \(l=5, w=4, h=3\)

\(V =\)

90.\(A = \frac{1}{2}bh\) when \(b = 12, h = 7\)

\(A =\)

91.\(F = \frac{9}{5}C + 32\) when \(C = 35\)

\(F =\)

92.\(C = \frac{5}{9}(F - 32)\) when \(F = 50\)

\(C =\)

93.\(3x - 4 = 11\). Is \(x = 5\) a sol?

Y / N

94.\(2y + 7 = 16\). Is \(y = 4\) a sol?

Y / N

95.\(4m + 5 = -7\). Is \(m = -3\) sol?

Y / N

96.\(\frac{k}{2} + 4 = 8\). Is \(k = 6\) sol?

Y / N

97.If \(3x + 4 = y\) and \(x = 6\), find \(y\)

\(y =\)

98.If \(a + 2b = 20\) and \(b = 7\), find \(a\)

\(a =\)

99.If \(\frac{x}{3} = 5\), what is \(x\)?

\(x =\)

100.\(\frac{x^2 - y^2}{x + y}\) when \(x = 9, y = 3\)

Ans:

Part 4 of 4 • Century Complete (100/100) 🎉 Congratulations on completing the Century Challenge!

-13

45. \((-3)^2\)

9

46. \((-5)^2\)

25

47. \(4-3(4)\)

-8

48. \(12+(-5)\)

7

49. \(-8+10\)

2

50. \(2+5+3\)

10

Diagnostic Flag: In #35, \(-x^2 + 10\) with \(x = 3\) means \(-(3^2) + 10 = -9 + 10 = 1\). Only the base is squared.

50 Problems Verified ✓

Century Master Key • Sheet 1 Page 2: Fractions (51–75) & Formulas (76–100) ➔

Teacher Master Key

Century Substitution Solutions • Problems 51–100

Page 2 of 2 (Q51–100)

Part 3: Fraction & Rational Intensive (51–75) Grouping in numerators: Simplify top before dividing

51. \(4 + 5\)

9

52. \(5 - 4\)

1

53. \(5 + 9\)

14

54. \(12 - 5\)

7

55. \(6 + 3\)

9

56. \(6 - 2\)

4

57. \(\frac{14}{2}\)

7

58. \(\frac{15}{3}\)

5

59. \(\frac{15}{5}\)

3

60. \(\frac{16}{4}\)

4

61. \(\frac{20}{2}\)

10

62. \(\frac{16}{4}\)

4

63. \(\frac{24}{3}\)

8

64. \(2 + 4\)

6

65. \(6 - 1\)

5

66. \(-3 + 6\)

3

67. \(-5 - 5\)

-10

68. \(\frac{6}{2}\)

3

69. \(\frac{-15}{5}\)

-3

70. \(\frac{-24}{6}\)

-4

71. \(12 + 2\)

14

72. \(\frac{50}{5}\)

10

73. \(\frac{22}{2}\)

11

74. \(5 + 4\)

9

75. \(\frac{5(36)}{9}\)

20

Part 4: Formulas, Geometry & Solutions (76–100) Verify units and logical true/false checks

76. \(10+12\)

22

77. \(24-10\)

14

78. \(24+7\)

31

79. \(40-9\)

31

80. \(9+16\)

25

81. \(64-36\)

28

82. \(3(10)\)

30

83. \(2(11)\)

22

84. \(3(9)\)

27

85. \(2 \times 3 \times 7\)

42

86. \(55 \times 4\)

220

87. \(15 \times 8\)

120

88. \(28+18\)

46

89. \(5 \times 4 \times 3\)

60

90. \(\frac{1}{2}(12)(7)\)

42

91. \(63+32\)

95

92. \(\frac{5}{9}(18)\)

10

93. \(11 = 11\)

YES

94. \(15 \neq 16\)

NO

95. \(-7 = -7\)

YES

96. \(7 \neq 8\)

NO

97. \(3(6)+4\)

22

98. \(20 - 14\)

6

99. \(5 \times 3\)

15

100. \(\frac{72}{12}\)

6

Fluency Scale:

90–100: Mastered • 80–89: Proficient • 70–79: Developing • <70: Review Fractions & Signs

100/100 Solutions Verified

Century Master Key • Sheet 2 End of 100-Problem Key