Testinar Grade 8 math

Estimating Expressions with Irrational Numbers

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

CH01-04 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.

On a number line, where would \(\sqrt{3}+1\) fall?
1. Between \(1\) and \(2\)
2. Greater than \(4\)
3. Between \(3\) and \(4\)
4. Between \(2\) and \(3\)
Which estimate has the largest error (furthest from the true value)?
1. \(\sqrt{9} = 3\)
2. \(\sqrt{10} \approx 3.2\)
3. \(\sqrt{11} \approx 3.3\)
4. \(\sqrt{12} \approx 3.6\)
A rectangle has dimensions \(2\) cm by \(3\) cm. The diagonal is \(\sqrt{13}\) cm. What is the best estimate of the diagonal?
1. \(3.0\) cm
2. \(6.5\) cm
3. \(4.2\) cm
4. \(3.6\) cm
Between which two consecutive integers does \(\sqrt{24}\) fall?
1. Between \(3\) and \(4\)
2. Between \(6\) and \(7\)
3. Between \(5\) and \(6\)
4. Between \(4\) and \(5\)
Which is the best estimate of \(\pi+\sqrt{10}\)?
1. \(3.1+3.1=6.2\)
2. \(\pi+10=13.14\)
3. \(4+4=8\)
4. \(3.14+3.16=6.30\)
Which is the best estimate of \(\sqrt{26}\)?
1. \(4.5\)
2. \(5.0\)
3. \(5.1\)
4. \(6.0\)
Which is the best estimate of \(\sqrt{2}\)?
1. \(1.2\)
2. \(2.8\)
3. \(2.0\)
4. \(1.4\)
Which is the best estimate of \(\pi\)?
1. \(2.14\)
2. \(3.0\)
3. \(3.14\)
4. \(3.5\)

Teacher and parent reference

Answer key and explanations

D. Between \(2\) and \(3\)
\(\sqrt{3}\approx1.73\), so \(\sqrt{3}+1\approx2.73\), which falls between \(2\) and \(3\). Distractors A, C, and D represent underestimation, overestimation, and far overestimation of \(\sqrt{3}\).
D. \(\sqrt{12} \approx 3.6\)
\(\sqrt{9}=3\) (exact), \(\sqrt{10}\approx3.162\) (error 0.04), \(\sqrt{11}\approx3.317\) (error 0.02), \(\sqrt{12}\approx3.464\) (error 0.14). Option D has error of 0.14, much larger than others.
D. \(3.6\) cm
By the Pythagorean theorem, diagonal \(= \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606 \approx 3.6\) cm.
D. Between \(4\) and \(5\)
\(4^2 = 16\) and \(5^2 = 25\). Since \(16 < 24 < 25\), we have \(4 < \sqrt{24} < 5\). More precisely, \(\sqrt{24} \approx 4.90\).
D. \(3.14+3.16=6.30\)
Using \(\pi\approx3.14\) and \(\sqrt{10}\approx3.16\) (since \(3.16^2\approx9.99\)), the sum is approximately \(6.30\).
C. \(5.1\)
Since \(5^2=25\) and \(5.1^2=26.01\approx26\), we have \(\sqrt{26}\approx5.1\). Distractors: 5.0 (close but low), 4.5 and 6.0 (too far).
D. \(1.4\)
Since \(1.4^2=1.96\approx2\) and \(1.5^2=2.25\), we have \(\sqrt{2}\approx1.41\).
C. \(3.14\)
\(\pi\approx3.14159\ldots\), commonly estimated as \(3.14\) or \(3.2\) for Grade 8 work. Distractors: A (underestimate), B (rounds down), D (rounds up).