Testinar Grade 8 math

Estimating Irrational Numbers

Learn to reason about and solve Grade 8 problems involving Estimating Irrational Numbers.

CH01-03 Number Systems and Financial Reasoning

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Where on a number line would \(\sqrt{45}\) be located?
1. Between \(5\) and \(6\)
2. Between \(4\) and \(5\)
3. Between \(6\) and \(7\)
4. At \(7\)
Where on a number line would \(\sqrt{77}\) be located?
1. At \(9\)
2. Between \(9\) and \(10\)
3. Between \(8\) and \(9\)
4. Between \(7\) and \(8\)
Using the table, between which integers does \(\sqrt{11}\) lie?
1. \(2\) and \(3\)
2. \(5\) and \(6\)
3. \(4\) and \(5\)
4. \(3\) and \(4\)
Which value belongs at the red mark on the number line?
1. \(\sqrt{9}\)
2. \(\sqrt{20}\)
3. \(\sqrt{16}\)
4. \(\sqrt{12}\)
Based on the number line, which decimal is closest to \(\sqrt{60}\)?
1. \(7.5\)
2. \(7.9\)
3. \(7.8\)
4. \(7.7\)
Is \(\sqrt{6}\) greater than \(2.4\) or less than \(2.4\)?
1. Cannot compare
2. Less than
3. Equal to
4. Greater than
Which list orders the values from least to greatest?
1. \(\sqrt{15}, \sqrt{5}, \sqrt[3]{8}\)
2. \(\sqrt{5}, \sqrt[3]{8}, \sqrt{15}\)
3. \(\sqrt{5}, \sqrt{15}, \sqrt[3]{8}\)
4. \(\sqrt[3]{8}, \sqrt{5}, \sqrt{15}\)
The diagonal of a square with side length \(1\) unit is \(\sqrt{2}\) units. Between which integers is this diagonal?
1. Between \(0\) and \(1\)
2. Between \(3\) and \(4\)
3. Between \(2\) and \(3\)
4. Between \(1\) and \(2\)

Teacher and parent reference

Answer key and explanations

C. Between \(6\) and \(7\)
\(6^2=36\) and \(7^2=49\). Since \(36<45<49\), we have \(6<\sqrt{45}<7\).
C. Between \(8\) and \(9\)
\(8^2=64\) and \(9^2=81\). Since \(64<77<81\), we have \(8<\sqrt{77}<9\).
D. \(3\) and \(4\)
From the table, \(3^2=9\) and \(4^2=16\). Since \(9<11<16\), we have \(3<\sqrt{11}<4\).
D. \(\sqrt{12}\)
The red mark is between \(3\) and \(4\). Since \(3^2=9\) and \(4^2=16\): \(\sqrt{9}=3\), \(\sqrt{16}=4\). Only \(\sqrt{12}\) satisfies \(9<12<16\), placing it between \(3\) and \(4\).
D. \(7.7\)
\(7.7^2=59.29\) and \(7.8^2=60.84\). The number \(60\) is a little closer to \(59.29\) than to \(60.84\), so \(\sqrt{60}\approx 7.746\) is closer to \(7.7\) than to \(7.8\).
D. Greater than
\((2.4)^2=5.76<6\), so \(2.4<\sqrt{6}\). Therefore \(\sqrt{6}\) is greater than \(2.4\).
D. \(\sqrt[3]{8}, \sqrt{5}, \sqrt{15}\)
Evaluate: \(\sqrt[3]{8}=2\), \(\sqrt{5}\approx 2.24\), \(\sqrt{15}\approx 3.87\). From the number line, order: \(2 < 2.24 < 3.87\).
D. Between \(1\) and \(2\)
Since \(1^2=1\) and \(2^2=4\), and \(1<2<4\), we have \(1<\sqrt{2}<2\).