Free beginning-of-year diagnostic

Algebra 1 Back-to-School Readiness Diagnostic

A 24-question Algebra 1 readiness diagnostic covering equations, slope, functions, exponents, radicals, systems, and geometric modeling.

24 questions6 skill areasAbout 70 minutesAnswer key and scoring guide
Download the free diagnostic PDF

What this diagnostic measures

Algebra 1 moves from isolated techniques to connected representations. Students need to maintain equality, interpret slope as a rate, evaluate functions, and use exponent and root notation without losing sight of what quantities mean.

Administration note

Require students to show one equation step per line and label coordinates clearly. A correct mental answer provides less diagnostic information than a visible method, especially for equations and linear models.

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.

Number and exponent fluency

Section score: ____ / 4

Use rational numbers, roots, and exponent rules.

  1. 1Evaluate -[[3^{2}]] + 4(-2).
  2. 2Simplify (2[[x^{3}]])(5[[x^{4}]]).
  3. 3Between which two integers is [[\sqrt{52}]]?
  4. 4Simplify [[\frac{12x^{5}}{3x^{2}}]], where [[x\neq0]].

Expressions and equations

Section score: ____ / 4

Simplify expressions and solve equations with variables on both sides.

  1. 5Simplify 4(3x - 2) - 5x + 1.
  2. 6Solve 6x - 11 = 2x + 17.
  3. 7Solve (x + 5)/4 = -3.
  4. 8Solve [[\frac{2x-3}{5}=7]].

Slope and lines

Section score: ____ / 4

Calculate slope and write linear equations.

  1. 9Find the slope through (-2, 7) and (4, -5).
  2. 10Write a line with slope 3 and y-intercept -4.
  3. 11Rewrite 2x + 4y = 12 in slope-intercept form.
  4. 12Write the equation of the line through [[(2,-1)]] and [[(6,7)]].

Functions

Section score: ____ / 4

Identify functions and connect tables to rules.

  1. 13Evaluate f(-2) for f(x) = 3[[x^{2}]] - 1.
  2. 14Is {(0, 2), (1, 5), (0, 7)} a function? Explain.
  3. 15A table has x-values 0, 2, 4 and y-values 1, 7, 13. Write its linear rule.
  4. 16For [[f(x)=x^{2}-4x+1]], find [[f(3)]].

Systems and geometry

Section score: ____ / 4

Solve a basic system and use coordinate geometry.

  1. 17Solve y = x + 2 and y = -2x + 11.
  2. 18Find the distance between (2, -1) and (8, 7).
  3. 19A right triangle has legs 8 and 15. Find the hypotenuse.
  4. 20Solve the system [[2x+y=11]] and [[x-y=1]].

Algebraic modeling

Section score: ____ / 4

Translate rates, constraints, and patterns into equations.

  1. 21Plan A costs $30 plus $5 per month. Plan B costs $12 plus $8 per month. When are they equal?
  2. 22A rectangle's length is twice its width plus 1. Its perimeter is 38. Find its dimensions.
  3. 23The sequence is 14, 9, 4, -1, ... Write a formula for the nth term.
  4. 24A taxi charges $[[4.50]] plus $[[2.25]] per mile. Write and solve an equation for a $[[22.50]] trip.
Open the answer key

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

Number and exponent fluency

1. -17

2. 10[[x^{7}]]

3. 7 and 8

4. [[4x^{3}]]

Expressions and equations

5. 7x - 7

6. x = 7

7. x = -17

8. [[x=19]]

Slope and lines

9. -2

10. y = 3x - 4

11. y = -1/2x + 3

12. [[y=2x-5]]

Functions

13. 11

14. No; input 0 has two outputs

15. y = 3x + 1

16. [[-2]]

Systems and geometry

17. (3, 5)

18. 10 units

19. 17

20. [[(4,3)]]

Algebraic modeling

21. After 6 months

22. Width 6, length 13

23. a_n = 19 - 5n

24. [[4.50+2.25m=22.50]]; [[m=8]] miles

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.

All back-to-school diagnostics8th Grade Back-to-School Math DiagnosticFull-length summer review packetsTarget one prerequisite skill