N.RN.3 Answers
1. D
8. Let a be rational, let b be irrational, and
let a + b = c. Assume c is rational.
Rewrite a + b = c as b = c + (a). Since a is
rational, −a is rational, and since the set
of rational numbers is closed under
addition, c + (−a) is rational. Thus, b is
rational. However, this contradicts the
condition that b is irrational. Thus, the
assumption that c is rational must be
invalid. This means that c, the sum of a
rational number and an irrational number,
must be irrational.
2. D
3. C
4. C, D
5. a.
b.
c.
d.
Irrational
Rational
Irrational
Irrational
(
)(
)
6. The expression 5 − 2 10 + 8 is
rational.
(
)(
Rubric
5 points for a logically correct argument
) (
)(
)
= ( 5 − 2 )(10 + 4 • 2 )
= ( 5 − 2 )(10 + 2 2 )
5 − 2 10 + 8 = 5 − 2 10 + 4 • 2
9. Let a ≠ 0 be rational, let b be irrational,
and let ab = c. Assume c is rational.
Rewrite ab = c as b = c •
= 50 + 10 2 − 10 2 − 4
= 46
46 is a rational number.
1
. Since a is
a
1
is rational, and
a
since the set of rational numbers is
1
is
closed under multiplication, c •
a
rational. Thus, b is rational. However, this
contradicts the assumption that b is
irrational. Thus, the assumption that c is
rational must be invalid. This means that
c, the product of a nonzero rational
number and an irrational number, must
be irrational.
rational and not zero,
Rubric
1 point for identifying the expression as
rational; 2 points for simplifying
(5 − 2 )(10 + 2 2 ) to 46 and noting that
46 is rational
7. Since the radius of the circle is rational, it
a
, where a
b
and b are nonzero integers. Then the
square of the radius of the circle can be
a a
written in the form
• , which is
b b
rational since the set of rational numbers
is closed under multiplication. The area of
the circle can then be written in the form
a a
π •
• . Since the area of the circle is
b b
the product of a nonzero rational number
and an irrational number, it must be
irrational.
can be written in the form
Rubric
5 points for a logically correct proof
Rubric
3 points for a logically correct argument
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Algebra 1 Teacher Guide
3
Common Core Assessment Readiness