Free beginning-of-year diagnostic

5th Grade Back-to-School Math Diagnostic

A free 24-question 5th grade back-to-school diagnostic assessing whole-number operations, factors, fractions, decimals, measurement, geometry, and modeling.

24 questions6 skill areasAbout 50 minutesAnswer key and scoring guide
Download the free diagnostic PDF

What this diagnostic measures

Fifth grade asks students to extend familiar operations to decimals and fractions. The most useful readiness evidence is not speed alone, but whether a student recognizes number relationships and selects an efficient operation in a mixed setting.

Administration note

Ask for an estimate beside each multi-digit computation. An implausible answer that a student accepts may signal a place-value need even when the calculation error itself is small.

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.

Whole-number fluency

Section score: ____ / 4

Compute and estimate with multi-digit whole numbers.

  1. 1Find 28,475 + 6,938.
  2. 2Find 50,000 - 18,746.
  3. 3Find 247 x 6.
  4. 4Estimate [[398\times21]] by rounding each factor to its greatest place.

Division and number structure

Section score: ____ / 4

Divide whole numbers and reason about factors and multiples.

  1. 5Find 1,764 [[\div]] 7.
  2. 6List every factor of 24.
  3. 7Is 91 prime or composite? Explain.
  4. 8Find the least common multiple of [[8]] and [[12]].

Fractions

Section score: ____ / 4

Compare fractions and combine fractions with like denominators.

  1. 9Use <, >, or =: [[\frac{5}{8}]] ___ [[\frac{2}{3}]].
  2. 10Find [[\frac{7}{12}]] + [[\frac{3}{12}]].
  3. 11Find 2 [[\frac{1}{5}]] - [[\frac{4}{5}]].
  4. 12Find [[\frac{3}{4}+\frac{1}{8}]].

Decimals and place value

Section score: ____ / 4

Connect fractions to decimals and interpret decimal places.

  1. 13Write [[\frac{37}{100}]] as a decimal.
  2. 14What is the value of the 6 in 12.064?
  3. 15Order from least to greatest: 0.7, 0.07, 0.707.
  4. 16Write [[0.45]] as a fraction in simplest form.

Measurement and geometry

Section score: ____ / 4

Convert units and use area, perimeter, and angle ideas.

  1. 17Convert 4 yards to feet.
  2. 18A rectangle has area 63 square inches and width 7 inches. Find its length.
  3. 19An angle measures 42 degrees. Is it acute, right, or obtuse?
  4. 20A rectangle is [[12]] cm long and [[5]] cm wide. Find both its perimeter and area.

Multi-step modeling

Section score: ____ / 4

Translate realistic situations into more than one operation.

  1. 21A school orders 18 boxes of 24 notebooks. It gives away 75. How many remain?
  2. 22Four friends split a $38.60 bill equally. How much does each pay?
  3. 23A trail is 3 [[\frac{1}{2}]] miles long. Priya walks 1 [[\frac{3}{4}]] miles. How much remains?
  4. 24A class packs [[6]] boxes with [[28]] pencils each, then shares the pencils equally among [[8]] tables. How many pencils per table?
Open the answer key

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

Whole-number fluency

1. 35,413

2. 31,254

3. 1,482

4. About 8,000

Division and number structure

5. [[252]]

6. 1, 2, 3, 4, 6, 8, 12, 24

7. Composite; 7 x 13 = 91

8. [[24]]

Fractions

9. <

10. [[\frac{10}{12}]], or [[\frac{5}{6}]]

11. 1 [[\frac{2}{5}]]

12. [[\frac{7}{8}]]

Decimals and place value

13. 0.37

14. 0.06, or six hundredths

15. 0.07, 0.7, 0.707

16. [[\frac{9}{20}]]

Measurement and geometry

17. 12 feet

18. 9 inches

19. Acute

20. Perimeter [[34]] cm; area [[60]] square cm

Multi-step modeling

21. 357 notebooks

22. $9.65

23. 1 [[\frac{3}{4}]] miles

24. [[21]] pencils

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 diagnostics4th Grade Back-to-School Math Diagnostic6th Grade Back-to-School Math DiagnosticFull-length summer review packetsTarget one prerequisite skill