Sums and Products of Rational and Irrational Numbers

5 min

Narrative

This activity prompts students to recognize that the sum of any two integers is always an integer and that the product of any two integers is also always an integer. Students will not be justifying in a formal way why these properties are true. In a future course, when studying polynomials, students will begin considering integers as a closed system under addition, subtraction, and multiplication. 

Some students may simply assert that these are true because they are not able to find two integers that add up to or multiply to make a noninteger. Others may reason that:

  • It seems impossible to add two whole amounts and end up with a partial amount in the sum.
  • It seems impossible to multiply whole-number groups of whole numbers to make a non-whole number quantity.
  • If we use the number line to represent addition of two integers, we will start at an integer and take whole-number steps, so we cannot end up at a point between integer values.

These are valid conclusions at this stage. Later in the lesson, students will use these conclusions to reason about the sums and products of rational numbers.

Launch

Arrange students in groups of 2. Ask them to think quietly for a couple of minutes before discussing their thinking with a partner.

Student Task

Here are some examples of integers:

  • -25
  • -10
  • -2
  • -1
  • 0
  • 5
  • 9
  • 40

  1. Experiment with adding any two numbers from the list (or other integers of your choice). Try to find one or more examples of two integers that:

    1. add up to another integer.
    2. add up to a number that is not an integer.

  2. Experiment with multiplying any two numbers from the list (or other integers of your choice). Try to find one or more examples of two integers that:

    1. multiply to make another integer.
    2. multiply to make a number that is not an integer.

Sample Response

  1. Sample response:
    1. 5+40=455 + 40 = 45, -10+9=-1\text-10 + 9 = \text-1
    2. no examples found
  2. Sample response:
    1. (5)(40)=200(5)(40) = 200, (-10)(9)=-90(\text-10)( 9) = \text-90
    2. no examples found

Synthesis

Select students or groups to share their examples and their challenges in finding examples for the second part of each question. Invite as many possible explanations as time permits for why they could not find examples of two integers that add or multiply to make a number that is not an integer. If no students bring up reasons similar to those listed in the Activity Narrative, ask students to consider them.

Explain that while we have not proven that two integers can never produce a sum or a product that is not an integer, for now we will accept this to be true.

Anticipated Misconceptions

Some students might persist in attempting to find examples of two integers that add up to a noninteger. Rather than giving them more time to find examples, encourage them to think about the placement of integers on a number line and what it might imply about the sum or product of any two integers.

Standards
Building On
  • 7.NS.1·Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
  • 7.NS.A.1·Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Building Toward
  • HSN-RN.B.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

15 min

15 min

30 min