This grade 8 irrational test answer key gives the full solution and the scoring guide for each of the 9 test questions on rational and irrational numbers.
Use this key to score the test out of 20 points. Each solution shows the expected work, and the scoring note shows how the points are earned. Give partial credit where a correct method has a small arithmetic slip.
Grade 8 Irrational Test Answer Key: questions 1–5
Question 1: Classify and justify (2 points)
- \(\sqrt{64}=8\), an integer, so rational.
- 27 is not a perfect square (\(5^{2}=25\), \(6^{2}=36\)), so \(\sqrt{27}\) is irrational.
Scoring: 1 point per part for the correct label with a valid reason.
Question 2: Fractions as decimals (2 points)
- \(9: 16=\)\(0.5625\).
- \(4: 15=\)\(0.2\overline{6}\) (the digit 6 repeats).
Scoring: 1 point per part. Accept 0.2667 only if the repeat is shown with a bar or dots.
Question 3: Repeating decimals to fractions (3 points)
- \(100x=18.1818\)… and \(x=0.1818\)…, so \(99x=18\) and \(x=\frac{18}{99}=\)\(\frac{2}{11}\).
- \(100x=43.33\)… and \(10x=4.33\)…. Subtract: \(90x=39\). So \(x=\frac{39}{90}=\)\(\frac{13}{30}\).
Scoring: (a) 1 point. (b) 1 point for a correct subtraction setup, 1 point for the simplified fraction.
Question 4: Square root of 75 (3 points)
- \(8^{2}=64\) and \(9^{2}=81\), so between 8 and 9.
- \(8.6^{2}=73.96\) and \(8.7^{2}=75.69\). 75 is closer to 75.69, so \(\sqrt{75}\approx\)\(8.7\).
Scoring: (a) 1 point. (b) 1 point for testing 8.6 and 8.7, 1 point for 8.7.
Question 5: Match points on a number line (2 points)
- \(3^{2}=9\) and \(4^{2}=16\), and \(3.2^{2}=10.24\). So \(\sqrt{10}\approx3.2\): point B.
- \(3.6^{2}=12.96\) is close to 13, so \(\sqrt{13}\approx3.6\): point C.
Scoring: 1 point per part for the correct point with a reasonable estimate.
Grade 8 Irrational Test Answer Key: questions 6–9
Question 6: Compare numbers (2 points)
- \(4.5^{2}=20.25\), which is greater than 20, so \(\sqrt{20}< 4.5\).
- \(\frac{22}{7}\approx3.1429\) and \(\pi\approx3.1416\), so \(\pi<\frac{22}{7}\).
Scoring: 1 point per part for the correct symbol with supporting work.
Question 7: Order from least to greatest (2 points)
\(\sqrt{5}\approx2.236\), \(\frac{9}{4}=2.25\), \(2.\overline{3}=2.333\)…. Order: \(2.2<\sqrt{5}<\frac{9}{4}< 2.\overline{3}\).
Scoring: 1 point for correct decimal values, 1 point for the correct order.
Question 8: Patio side length (2 points)
The side is \(\sqrt{55}\). \(7^{2}=49\) and \(8^{2}=64\). Then \(7.4^{2}=54.76\) and \(7.5^{2}=56.25\). 55 is closer to 54.76, so the side is about \(7.4\text{ ft}\).
Scoring: 1 point for bracketing the root between two decimals, 1 point for 7.4 feet.
Question 9: Pattern decimal (2 points)
The digits do not repeat a fixed block, because the number of 2s keeps growing. The decimal never ends and never repeats, so it is irrational. Sofia is not correct.
Scoring: 1 point for saying irrational, 1 point for the reason (no repeating block).
Scoring summary
20 points: add the points of every question of this grade 8 irrational test answer key.
How to use this grade 8 irrational test answer key
Use this grade 8 irrational test answer key to score the test fairly and to see which skills need more practice.
Printable PDF · Test Answer Key
Grade 8 Rational and Irrational Numbers test answer key
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