Free beginning-of-year diagnostic

8th Grade Back-to-School Math Diagnostic

A free 24-question 8th grade back-to-school diagnostic on rational numbers, proportions, percent, equations, geometry, probability, and linear relationships.

24 questions6 skill areasAbout 65 minutesAnswer key and scoring guide
Download the free diagnostic PDF

What this diagnostic measures

Eighth grade connects proportional thinking to slope, functions, systems, and exponents. This diagnostic checks the seventh-grade foundations that let students focus on those new connections instead of backtracking through arithmetic.

Administration note

Encourage equations for word problems even when mental reasoning seems faster. The ability to represent a situation symbolically is a central readiness skill for eighth-grade algebra.

Preview all 24 questions

Give one point per correct answer. Also record each section score out of three so the follow-up plan stays specific.

Signed rational numbers

Section score: ____ / 4

Compute with positive and negative fractions and decimals.

  1. 1Find -3.8 + 6.25.
  2. 2Find ([[\frac{-4}{9}]])([[\frac{27}{8}]]).
  3. 3Find [[(-15\div2.5)-4]].
  4. 4Evaluate [[\frac{5}{6}-\left(-\frac{3}{4}\right)]].

Proportions and percent

Section score: ____ / 4

Solve proportions, scale problems, and percent change.

  1. 5Solve [[\frac{9}{14}]] = x/42.
  2. 6A 3-inch map distance represents 45 miles. What does 7 inches represent?
  3. 7A price rises from $80 to $92. Find the percent increase.
  4. 8A sweater costs $[[72]] after a [[20\%]] discount. Find the original price.

Expressions

Section score: ____ / 4

Apply properties and factor using the greatest common factor.

  1. 9Simplify 3(4x - 5) - 2x.
  2. 10Factor 24x + 36 completely.
  3. 11Evaluate [[2x^{2}-3y]] when [[x=-3]] and [[y=4]].
  4. 12Simplify [[5(2x-3)-4(x+1)]].

Equations

Section score: ____ / 4

Solve multi-step equations and inequalities.

  1. 13Solve 5x - 8 = 2x + 19.
  2. 14Solve -4(x - 3) = 28.
  3. 15Solve 3x + 7 < 22.
  4. 16Solve [[-2(3x-1)\geq20]].

Geometry and probability

Section score: ____ / 4

Use circle formulas, angle relationships, and probability.

  1. 17Find the circumference of a circle with radius 5. Use [[\pi]] = 3.14.
  2. 18Two supplementary angles measure (2x + 10) degrees and (4x - 4) degrees. Find x.
  3. 19A bag has 3 red, 4 blue, and 5 green tiles. Find P(not green).
  4. 20A right triangle has legs [[9]] and [[12]]. Find the hypotenuse.

Linear readiness

Section score: ____ / 4

Recognize constant rates and build linear models.

  1. 21The points are (1, 4), (3, 10), and (5, 16). Find the constant rate of change.
  2. 22A service costs $18 plus $6 per hour. Write an equation for cost C after h hours.
  3. 23The perimeter of a rectangle is 46. Its length is 5 more than its width. Find both dimensions.
  4. 24For [[y=-\frac{3}{2}x+7]], identify the slope and [[y]]-intercept.
Open the answer key

Equivalent forms and clearly justified open-ended responses should receive credit.

Signed rational numbers

1. 2.45

2. [[\frac{-3}{2}]], or -1.5

3. [[-10]]

4. [[\frac{19}{12}]]

Proportions and percent

5. x = 27

6. 105 miles

7. [[15\%]]

8. $[[90]]

Expressions

9. 10x - 15

10. 12(2x + 3)

11. [[6]]

12. [[6x-19]]

Equations

13. x = 9

14. x = -4

15. x < 5

16. [[xleq-3]]

Geometry and probability

17. 31.4 units

18. x = 29

19. [[\frac{7}{12}]]

20. [[15]]

Linear readiness

21. 3

22. C = 6h + 18

23. Width 9, length 14

24. Slope [[-\frac{3}{2}]]; [[y]]-intercept [[7]]

Readiness score

Turn the answer key into a next step.

Count one point for each correct answer. Partial understanding is best noted beside the problem rather than counted as half a point.

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