Comparing Rational Expressions

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Comparing Rational Expressions

Student Outcomes

  • Students compare rational expressions by writing them in different but equivalent forms.

New York State Common Core Math Algebra II, Module 1, Lesson 23

Worksheets for Algebra II, Module 1, Lesson 23


Opening Exercise Use the slips of paper you have been given to create visual arguments for whether 1/3 or 3/8 is larger.


We will start by working with positive integers. Let 𝑚 and 𝑛 be positive integers. Work through the following exercises with a partner.

  1. Fill out the following table.
  2. Do you expect 1/𝑛 to be larger or smaller than 1/𝑛+1? Do you expect 1/𝑛 to be larger or smaller than 1/𝑛+2? Explain why.
  3. Compare the rational expressions 1/𝑛, 1/(𝑛+1), and 1/(𝑛+2) for 𝑛 = 1, 2, and 3. Do your results support your conjecture from Exercise 2? Revise your conjecture if necessary.
  4. From your work in Exercises 1 and 2, generalize how 1/𝑛 compares to 1/(𝑛+𝑚), where 𝑚 and 𝑛 are positive integers.
  5. Will your conjecture change or stay the same if the numerator is 2 instead of 1? Make a conjecture about what happens when the numerator is held constant, but the denominator increases for positive numbers.

Lesson Summary

To compare two rational expressions, find equivalent rational expression with the same denominator. Then we can compare the numerators for values of the variable that do not cause the rational expression to change from positive to negative or vice versa.

We may also use numerical and graphical analysis to help understand the relative sizes of expressions.

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