Rational and Irrational Numbers: Grade 8 Math Test Answer Key – PDF Download

Question 1: Classify and justify (2 points)

1. \(\sqrt{64}=8\), an integer, so rational.
2. 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)

1. \(9: 16=\)\(0.5625\).
2. \(4: 15=\)\(0.2\overline{6}\) (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)

1. \(100x=18.1818\)… and \(x=0.1818\)…, so \(99x=18\) and \(x=\frac{18}{99}=\)\(\frac{2}{11}\).
2. \(100x=43.33\)… 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)

1. \(8^{2}=64\) and \(9^{2}=81\), so between 8 and 9.
2. \(8.6^{2}=73.96\) 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)

1. \(3^{2}=9\) and \(4^{2}=16\), and \(3.2^{2}=10.24\). So \(\sqrt{10}\approx3.2\): point B.
2. \(3.6^{2}=12.96\) is close to 13, so \(\sqrt{13}\approx3.6\): point C.

Scoring: 1 point per part for the correct point with a reasonable estimate.

Question 6: Compare numbers (2 points)

1. \(4.5^{2}=20.25\), which is greater than 20, so \(\sqrt{20}< 4.5\).
2. \(\frac{22}{7}\approx3.1429\) 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).

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