VTCAP Grade 8 The Number System. Practice it free.

Grade 8 Number System standards; reported jointly with expressions and equations as 'The Number System, Expressions and Equations' performance indicator. Blueprint target: 4 points (7% of the grade 8 test). This Grade 8 reporting domain maps to 1 practice skill and 8 representative questions from the playable bank.

Grade 88.NS1 mapped skills
What the test measures

The Number System skills

  1. Know that there are numbers that are not rational, and approximate them by rational numbers

Standards basis

Common Core State Standards for Mathematics (CCSS-M) — Vermont

How the VTCAP reports it

Vermont administered the Smarter Balanced summative through spring 2022, then replaced it with the Cognia-developed Vermont Comprehensive Assessment Program (VTCAP) beginning spring 2023 — this landing slug predates the switch. VTCAP mathematics is an adaptive, CCSS-M-aligned assessment in grades 3-9. It is NOT a Smarter Balanced administration: its reporting categories are groupings of the CCSS grade-level domains (e.g. 'The Number System, Expressions and Equations'), not SBAC claims, though two cross-grade Mathematical Practices indicators ('Problem Solving, Reasoning, and Argument'; 'Modeling, Patterns, and Structure') echo the old claim structure. Per the test specifications, the grades 3-8 test may assess any CCSS-M standard at the grade; the grade 9 test assesses a prioritized subset of the high school CCSS-M within the conceptual categories of Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability (the assessed focus areas are published in the grade 9 Performance Level Descriptors).

Blueprint & weighting

VTCAP Test Specifications (Spring 2026) operational test blueprint, target percent of points per grade — Grade 3: OA 24-35%, NBT 10-14%, NF 16-20%, MD 22-29%, G 6-10%, constructed-response Mathematical Practices 12%. Grade 4: OA 20-31%, NBT 16-20%, NF 20-31%, MD 12-20%, G 6-10%, practices 12%. Grade 5: OA 14-22%, NBT 14-25%, NF 22-29%, MD 20-27%, G 8-16%, practices 12%. Grade 6: RP 15-22%, NS 15-22%, EE 15-22%, G 11-19%, SP 11-19%, practices 11%. Grade 7: RP 15-22%, NS 11%, EE 15-30%, G 11-19%, SP 19-22%, practices 11%. Grade 8: F 18-29%, NS 7%, EE 20-31%, G 18-29%, SP 18-22%, practices 11%. Grade 9 (all machine-scored): Number and Quantity/Algebra 18-27%, Algebra/Functions 31-40%, Geometry 20-29%, Statistics and Probability 13-20%; the two practices indicators are dual-aligned targets of >18% of points each. Identical figures appear in the Spring 2025 specifications.

Practice by skill

Topics mapped to this domain

Irrational number approximations

Grade 820 questions
Try it now

8 free questions · 0/0 correct

Practice playeasy

27\sqrt{27} lies between two consecutive integers. Enter the larger one.

52=25<27<36=625^2 = 25 < 27 < 36 = 6^2, so 27\sqrt{27} is between 55 and 66; the larger is 66.

Practice playeasy

What integer is closest to 120\sqrt{120}?

112=12111^2 = 121 sits right above 120120 and 120≈10.95\sqrt{120} \approx 10.95, so the closest integer is 1111.

Practice playeasy

What is the integer part of 10+2\sqrt{10} + 2?

10≈3.16\sqrt{10} \approx 3.16, so 10+2≈5.16\sqrt{10} + 2 \approx 5.16; the integer part is 55.

Practice playeasy

Which is closest to π\pi?

π≈3.14159\pi \approx 3.14159.

Practice playeasy

Which is closest to 2\sqrt{2}?

2≈1.414\sqrt{2} \approx 1.414.

Practice playeasy

Which is closest to 3\sqrt{3}?

3≈1.732\sqrt{3} \approx 1.732.

Practice playeasy

Which is closest to 5\sqrt{5}?

5≈2.236\sqrt{5} \approx 2.236.

Practice playeasy

40\sqrt{40} is between two consecutive integers. Enter the smaller one.

62=36<40<49=726^2 = 36 < 40 < 49 = 7^2, so 40\sqrt{40} is between 66 and 77; the smaller is 66.

Keep practicing

Turn the number system into game time.

The VTCAP placement starts with this test's real coverage map and finds the right difficulty.

Batting Cage game coverBatting CagePlay →Blaster Gate game coverBlaster GatePlay →Blitz Tanks game coverBlitz TanksPlay →Buzzer Beater game coverBuzzer BeaterPlay →
Related Common Core standards
8.NS.A.2
The Number System in other VTCAP grades
Grade 6Grade 7
The same skills on other tests
i-ReadyAASAACAPATLAS
Source and test overview
All VTCAP prepOfficial test source
Reporting categories

More Grade 8 domains

FunctionsGrade 8 Functions standards; reported as the 'Functions' performance indicator. Blueprint target: 10-16 points (18-29% of the grade 8 test).8 QsPractice →Expressions and EquationsGrade 8 Expressions and Equations standards; reported jointly with the number system as 'The Number System, Expressions and Equations' performance indicator. Blueprint target: 11-17 points (20-31% of the grade 8 test).8 QsPractice →GeometryGrade 8 Geometry standards; reported jointly with statistics as the 'Geometry and Statistics and Probability' performance indicator. Blueprint target: 10-16 points (18-29% of the grade 8 test).8 QsPractice →Statistics and ProbabilityGrade 8 Statistics and Probability standards; reported jointly with geometry as the 'Geometry and Statistics and Probability' performance indicator. Blueprint target: 10-12 points (18-22% of the grade 8 test).8 QsPractice →