Task Sets: Explain operations on rational and irrational numbers - Common Core Math - HS Number and Quantity

Warm-Up: Classifying Real Numbers

Classify each number and explain your reasoning. Simplify any radicals before classifying.

Which of the following is an irrational number?

$\sqrt{25}$

$0.121212\ldots$

$\sqrt{7}$

$-\frac{3}{4}$

Which of the following is rational?

$\sqrt{11}$

$\sqrt{36}$

$\sqrt{2}$

$\pi$

State the definition of a rational number. Then explain what makes a number irrational. Give one example of each.

Fluency Practice

For each problem, classify the expression and provide the required argument or proof.

Classify $5 + \sqrt{9}$ . (Simplify the radical first.)

Irrational, because it is a sum involving a square root.

Rational, because $\sqrt{9} = 3$ and $5 + 3 = 8$ , which is rational.

Irrational, because adding 5 to a square root always gives an irrational number.

Rational, but only because the rational part (5) is a whole number.

Use the algebraic construction to explain why the sum of two rational numbers is always rational. Your argument must work for ALL pairs of rational numbers, not just a specific example. Let $r = \frac{p}{q}$ and $s = \frac{m}{n}$ be any two rational numbers.

Prove by contradiction that $4 + \sqrt{3}$ is irrational. Use the proof-by-contradiction template: (1) state what you want to prove, (2) assume the OPPOSITE, (3) derive a contradiction, (4) state your conclusion.

Using the same proof-by-contradiction template, prove that $\frac{1}{2} + \sqrt{5}$ is irrational.

Prove by contradiction that $3 \cdot \sqrt{2}$ is irrational. Then explain specifically why the proof requires the factor 3 to be nonzero — what goes wrong if we try to apply the same argument to $0 \cdot \sqrt{2}$ ?

Exercises: Explain Why Sums and Products of Rational and Irrational Numbers Follow Specific Rules

Warm-Up: Classifying Real Numbers

Fluency Practice