From rational numbers to the real number line
• Work with infinite decimal expansions of numbers on the number line.
• Reason about operations with rational and irrational numbers (N-RN.B.3).
In this section, students take a deeper look at the number line. In elementary school they learned how to place fractions on the number line and in middle school they added negative numbers and learned about the existence of a few isolated irrational numbers, such as $\pi$ and $\sqrt{2}$. Now they see infinite decimal expansions as a way of locating any number, rational or irrational, on the number line. They also expand their repertoire of irrational numbers by considering sums and products of rational with irrational numbers.
Tasks
1
Identifying Rational Numbers
WHAT: These are two 8th grade tasks focused on converting repeating decimals to fractions and identifying rational and irrational numbers. The only two irrational numbers in this activity are π and $\sqrt2$.
WHY: These tasks motivate the later work in the section. Although it is simple to tell if a number is rational by expressing it as a quotient of integers, it is difficult in most cases to be sure that a number is irrational. Students are expected to know that π and are irrational. Proving that π is irrational is not within the scope of high school mathematics. In the next activity, students see a partial proof that is irrational, and in the activity after that they see how to construct more irrational numbers from known ones through the operations of addition and subtraction.
2
Calculating the square root of 2
WHAT: In this task students see why the square of a terminating decimal can never be equal to 2, and that the calculator’s result must be an approximation. Since there are rational numbers with non-terminating decimals this does not quite prove that is irrational, but it is a step in that direction and the argument can be generalized to give a complete proof (MP.3).
WHY: This task goes beyond the work with infinite decimal expansions that students did in Grade 8. A calculator will always report the square root of 2 as a terminating decimal because it only has a finite number of places it can report. This task gives students insight into why they must regard calculator answers as approximations, and helps them appreciate the mysterious nature of real numbers expressed as infinite decimal expansions (MP.5).
3
Rational or Irrational?
WHAT: This task builds on Activity 1.1 by providing students opportunities to reason about irrational numbers in various ways, including explaining why a sum or a product of a rational a number and an irrational number is irrational. In addition to eliciting several different types of reasoning (MP.2), the task requires students to rewrite radical expressions in which the radicand is divisible by a perfect square.
WHY: The solutions to this task are written as formal arguments; teachers are encouraged to engage students in a dialogue (or have them engage each other in groups) to help them develop rigorous arguments for the rationality and irrationality of each of the given numbers (MP.3).