Comparison of Irrational Numbers

Examples, solutions, and videos to help Grade 8 students learn how to use rational approximations of irrational numbers to compare the size of irrational numbers and place irrational numbers in their approximate locations on a number line.

New York State Common Core Math Grade 8, Module 7, Lesson 13

Download Worksheets for Grade 8, Module 7, Lesson 13

Lesson 13 Student Outcomes

• Students use rational approximations of irrational numbers to compare the size of irrational numbers.
• Students place irrational numbers in their approximate locations on a number line.

Lesson 13 Summary

The decimal expansion of rational numbers can be found by using long division, equivalent fractions, or the method of rational approximation.
The decimal expansion of irrational numbers can be found using the method of rational approximation.

Lesson 13 Classwork

Opening Exercise
Exercises 1–11

1. Rodney thinks that \(\sqrt[3]{{64}}\) is greater than \(\frac{{17}}{4}\). Sam thinks that \(\frac{{17}}{4}\) is greater. Who is right and why?
2. Which number is smaller, \(\sqrt[3]{{27}}\) or 2.89? Explain
3. Which number is smaller, \(\sqrt{{121}}\) or \(\sqrt[3]{{125}}\)? Explain.
4. Which number is smaller, \(\sqrt{{49}}\) or \(\sqrt[3]{{216}}\)? Explain.
5. Which number is greater, \(\sqrt{{50}}\) or \(\frac{{319}}{45}\)? Explain.
6. Which number is greater, \(\frac{{5}}{11}\) or \(0.\overline 4 \)? Explain.
7. Which number is greater, \(\sqrt{{38}}\) or \(\frac{{154}}{25}\)? Explain.
8. Which number is greater, \(\sqrt{{2}}\) or \(\frac{{15}}{9}\)? Explain.
9. Place the following numbers at their approximate location on the number line: \(\sqrt{25}\),\(\sqrt{28}\),\(\sqrt{30}\),\(\sqrt{32}\),\(\sqrt{35}\),\(\sqrt{36}\)
10. Challenge: Which number is larger \(\sqrt{5}\) or \(\sqrt[3]{11}\)?
11. A certain chessboard is being designed so that each square has an area of 3in². What is the length, rounded to the tenths place, of one edge of the board? (A chessboard is composed of 64 squares as shown.)

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