8th Grade Rational and Irrational Numbers Test Solutions and Scoring (PDF)

1. \(7 \div 8 = 0.875\).
2. \(4 \div 9 = 0.444…\), so \(0.\overline{4}\).
3. Yes. It is a fraction of two integers (its decimal terminates).

Scoring: 1 point for 0.875. 1 point for \(0.\overline{4}\). 1 point for a correct reason.

1. \(10x – x = 5\), so \(x = \frac{5}{9}\).
2. \(100x – x = 18\), so \(99x = 18\) and \(x = \frac{18}{99} = \frac{2}{11}\).

Scoring: Part a: 1 point. Part b: 1 point for the setup \(99x = 18\), 1 point for \(\frac{2}{11}\).

\(5^{2} = 25\) and \(6^{2} = 36\), so \(5 < \sqrt{31} < 6\).

\(5.5^{2} = 30.25\) and \(5.6^{2} = 31.36\); 31 is closer to 31.36.

5 and 6; about 5.6

Scoring: 1 point for 5 and 6. 1 point for 5.6 with work.

1. rational (equals 9)
2. irrational (15 is not a perfect square)
3. rational (repeating decimal)
4. irrational (a nonzero rational times \(\pi\))

Scoring: Half a point each, rounded up: 2 points for 4 correct, 1 point for 2 or 3 correct.

\(\sqrt{10} \approx 3.162\), \(\frac{16}{5} = 3.2\), \(3.\overline{2} = 3.2222…\)

\(3.1,\ \sqrt{10},\ \frac{16}{5},\ 3.\overline{2}\)

Scoring: 2 points for the correct order; 1 point if only one number is misplaced.

1. \(8.7^{2} = 75.69 > 75\), so \(\sqrt{75} < 8.7\).
2. \(\pi = 3.14159… > 3.14\).

Scoring: 1 point for each correct comparison with a reason.

1. Side \(= \sqrt{90}\).
2. \(9^{2} = 81\) and \(10^{2} = 100\), so between 9 and 10.
3. \(9.5^{2} = 90.25\) and \(9.4^{2} = 88.36\); 90 is closer to 90.25: about 9.5 ft.

Scoring: 1 point per part.

Let \(x = 0.\overline{9}\). Then \(10x = 9.\overline{9}\), and \(9x = 9\), so \(x = 1\).

Maya is wrong: 0.\overline{9} equals 1, a rational number.

Scoring: 1 point for saying she is wrong. 1 point for setting up and subtracting. 1 point for the result \(x = 1\).