Lesson 21

Sums and Products of Rational and Irrational Numbers

  • Let’s make convincing arguments about why the sums and products of rational and irrational numbers always produce certain kinds of numbers.

Problem 1

Match each expression to an equivalent expression.

(From Unit 7, Lesson 15.)

Problem 2

Consider the statement: "An irrational number multiplied by an irrational number always makes an irrational product."

Select all the examples that show that this statement is false.

A:

\(\sqrt4\boldcdot\sqrt5\)

B:

\(\sqrt4\boldcdot\sqrt4\)

C:

\(\sqrt7\boldcdot\sqrt7\)

D:

\(\frac{1}{\sqrt5}\boldcdot\sqrt5\)

E:

\(\sqrt0\boldcdot\sqrt7\)

F:

\(\text-\sqrt5\boldcdot\sqrt5\)

G:

\(\sqrt5\boldcdot\sqrt7\)

Problem 3

  1. Where is the vertex of the graph that represents \(y=(x-3)^2 + 5\)?
  2. Does the graph open up or down? Explain how you know.
(From Unit 6, Lesson 15.)

Problem 4

Here are the solutions to some quadratic equations. Decide if the solutions are rational or irrational.

\(3 \pm \sqrt2\)

\(\sqrt9 \pm 1\)

\(\frac12 \pm \frac32\)

\(10 \pm 0.3\)

\(\frac{1 \pm \sqrt8}{2} \)

\(\text-7\pm\sqrt{\frac49}\)

 

 

Problem 5

Find an example that shows that the statement is false.

  1. An irrational number multiplied by an irrational number always makes an irrational product.
  2. A rational number multiplied by an irrational number never gives a rational product.
  3. Adding an irrational number to an irrational number always gives an irrational sum.

Problem 6

Which equation is equivalent to \(x^2-\frac32x=\frac74\) but has a perfect square on one side?

A:

\(x^2-\frac32x+3=\frac{19}{4}\)

B:

\(x^2-\frac32x+\frac34=\frac{10}{4}\)

C:

\(x^2-\frac32x+\frac94=\frac{16}{4}\)

D:

\(x^2-\frac32x+\frac94=\frac74\)

(From Unit 7, Lesson 13.)

Problem 7

A student who used the quadratic formula to solve \(2x^2-8x=2\) said that the solutions are \(x=2+\sqrt5\) and \(x=2-\sqrt5\)

  1. What equations can we graph to check those solutions? What features of the graph do we analyze?
  2. How do we look for \(2+\sqrt5\) and \(2-\sqrt5\) on a graph?
(From Unit 7, Lesson 18.)

Problem 8

Here are 4 graphs. Match each graph with a quadratic equation that it represents.

Graph A

A parabola in x y plane, origin O. X axis negative 8 to 6, by 2’s. Y axis negative 6 to 4, by 2s. Opens upward with vertex at 4 comma 3.

Graph B

Parabola. Opens up. Vertex = 4 comma -3.

Graph C

Parabola. Opens up. Vertex = -4 comma 3.

Graph D

Parabola. Opens up. Vertex = -4 comma -3.
(From Unit 6, Lesson 15.)