Grade 8
Name___Answer Key____
Numbers and Operations
Determine if the given number is rational or irrational (circle your answer). Explain your
thinking.
Number
1a
Circle your answer
Rational or Irrational
This number is not a perfect square which means the number is
irrational because it cannot be written as a terminating or repeating
decimal.
Rational or Irrational
This number is rational because it can be written as a terminating
9
decimal. It can also be written as 10 which is defined to be a rational
number.
Rational or Irrational
This number is rational because it can be written as a terminating
1
decimal (.2). it can is written as 5 which is defined to be a rational
number.
√5
1b
0.9
1c
1
5
1d
1e
Rational or Irrational
𝜋
√9
Explain your thinking.
Rational or Irrational
This number is a non- terminating or non- repeating decimal which means it is
irrational.
This number is a perfect square which means the number is rational.
√9 is equal to 3 (the square root symbol indicates the principal root only). 3
3
can be written as 1 which is a rational number by definition.
2. Place the following numbers on the number line: -1.4,
Explain your thinking for each number’s placement.
Sample Response:
, and
Grade 8
Numbers and Operations
Name___Answer Key____
-1.4 is between -2 and -1 (closer to -1 because it is greater than -1.5 which would be in the
middle between -1 and -2.
is between √1 and √4. √1 = 1 𝑎𝑛𝑑 √4 = 2. Also √2 is closer to √1 than √4 on the number line.
√2 ≈ 1.4142
is greater than 2.2, but less than 2.3 (and closer to 2.2 than 2.3).
Rational Numbers: Any number that can be written in fraction form is a rational number. This
includes integers, terminating decimals, and repeating decimals as well as fractions.
•
•
An integer can be written as a fraction simply by giving it a denominator of one, so any
integer is a rational number.
;
;
A terminating decimal can be written as a fraction simply by writing it the way you say
it: 3.75 = three and seventy-five hundredths =
•
, then adding if needed to produce a
fraction:
. So, any terminating decimal is a rational number.
A repeating decimal can be written as a fraction using algebraic methods, so any
repeating decimal is a rational number.
An irrational number is any number that can’t be written in fraction form (also include nonterminating and non-repeating decimals).
Grade 8
Numbers and Operations
Name___Answer Key____
Scoring Guide
1
2
•
No answer given or
only 1 answer is
correct.
•
Correct response is
given for 2 or 3
numbers.
•
Correct response is
given for 4 or 5
numbers.
•
No explanation or
explanation is poorly
written.
•
Explanation is wellwritten, but lacks solid
mathematical
evidence for the
response.
•
•
No answer given or all
numbers are
incorrectly placed.
•
Correct placement
(approximately) is
given for 1 or 2
numbers.
•
Explanation is well
written and
mathematical
evidence from the
data is connected to
the response.
Correct placement
(approximately) is
given for all 3
numbers.
•
No explanation or
explanation is poorly
written.
•
Explanation is wellwritten, but lacks solid
mathematical
evidence for the
response.
•
Question 1
Question 1 Explanation
Question 2
Question 2 Explanation
0
Explanation is well
written and
mathematical
evidence from the
data is connected to
the response.
Grade 8
Numbers and Operations
Name___Answer Key____
Suggested answers and possible explanations.
Number
1a
1b
Circle your answer
Rational or Irrational
√5
Rational or Irrational
Explain your thinking.
Non-repeating, non-terminating decimal number.
Terminating decimal number.
0.9
1c
Rational or Irrational
1d
1e
1
5
𝜋
√9
Rational or Irrational
Rational or Irrational
Terminating decimal number.
Non-repeating, non-terminating decimal number.
Principle root is 3 which is a rational number.