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 are always certain kinds of numbers.
Problem 1
Match each expression to an equivalent expression.
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.
\(\sqrt4\boldcdot\sqrt5\)
\(\sqrt4\boldcdot\sqrt4\)
\(\sqrt7\boldcdot\sqrt7\)
\(\frac{1}{\sqrt5}\boldcdot\sqrt5\)
\(\sqrt0\boldcdot\sqrt7\)
\(\text-\sqrt5\boldcdot\sqrt5\)
\(\sqrt5\boldcdot\sqrt7\)
Problem 3
- Where is the vertex of the graph that represents \(y=(x-3)^2 + 5\)?
- Does the graph open up or down? Explain how you know.
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.
- An irrational number multiplied by an irrational number always makes an irrational product.
- A rational number multiplied by an irrational number never gives a rational product.
- Adding an irrational number to an irrational number always gives an irrational sum.
Problem 6
Which equation is equivalent to \(x^2-3x=\frac74\) but has a perfect square on one side?
\(x^2-3x+3=\frac{19}{4}\)
\(x^2-3x+\frac34=\frac{10}{4}\)
\(x^2-3x+\frac94=\frac{16}{4}\)
\(x^2-3x+\frac94=\frac74\)
Problem 7
A student who used the quadratic formula to solve \(2x^2-8x=2\) said that the solutions are \(x=2+\sqrt5\) and \(x=2-\sqrt5\).
- What equations can we graph to check those solutions? What features of the graph do we analyze?
- How do we look for \(2+\sqrt5\) and \(2-\sqrt5\) on a graph?
Problem 8
Here are 4 graphs. Match each graph with a quadratic equation that it represents.
Graph A
Graph B
Graph C
Graph D
