Math
Released Set 2015
Algebra 1
PBA Item #13
Two Real Numbers Defined
M44105
Prompt
Rubric
Task is worth a total of 3 points.
M44105 Rubric
Score Description
3
Student response includes the following 3 elements.
• Reasoning component = 3 points
o Correct identification of a as rational and b as irrational
o Correct identification that the product is irrational
o Correct reasoning used to determine rational and
irrational numbers
Sample Student Response:
A rational number can be written as a ratio. In other words, a
number that can be written as a simple fraction.
a = 0.444444444444... can be written as 4 . Thus, a is a
9
rational number. All numbers that are not rational are considered
irrational. An irrational number can be written as a decimal, but
not as a fraction. b = 0.354355435554... cannot be written as a
fraction, so it is irrational. The product of an irrational number
and a nonzero rational number is always irrational, so the
product of a and b is irrational. You can also see it is irrational
with my calculations:
4 (.354355435554...) = .15749...
9
2
.15749... is irrational.
Student response includes 2 of the 3 elements.
1
0
Student response includes 1 of the 3 elements.
Student response is incorrect or irrelevant.
Anchor Set
A1 – A8
A1
Score Point 3
Annotations
Anchor Paper 1
Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
•
•
Correct identification of a as rational and b as irrational (The number represented by
a is rational . . . The number represented by b would be irrational).
Correct identification that the product is irrational (The product of the expression ab
would equal an irrational number).
Correct reasoning to determine rational and irrational numbers (rational because it
repeats, and can be written as a fraction), (irrational . . . it cannot be written as a
fraction), (a nonzero rational number multiplied by an irrational number can never
be rational, only irrational . . . cannot be written as a fraction).
Note: Since a is defined as a = 0.444… we know that the product will not be zero, so
we know the student’s explanation is sufficient.
A2
Score Point 3
Annotations
Anchor Paper 2
Score Point 3
This response receives full credit. The student includes each of the three required elements:
•
•
•
Correct identification of a as rational and b as irrational (a is rational . . . b is
irrational).
Correct identification that the product is irrational (the product of a and b is
irrational).
Correct reasoning to determine rational and irrational numbers (rational because it is
repeated . . . irrational because the digits have no specific pattern).
A3
Score Point 2
Annotations
Anchor Paper 3
Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
•
Correct identification of a as rational and b as irrational (a = rational), (b = irrational).
Correct identification that the product is irrational (The product of a and b would have
to be irrational).
The student does not provide reasoning to determine rational and irrational numbers.
A4
Score Point 2
Annotations
Anchor Paper 4
Score Point 2
This response receives partial credit. The student includes two of the three required
elements:
•
•
Correct identification of a as rational and b as irrational (a is rational . . . b is not
rational).
Correct reasoning to determine rational and irrational numbers (rational because it
can be expressed as a fraction ( 94 ) . . . not rational because it can’t be expressed as a
fraction).
The student does not indicate that the product of the two numbers would be irrational.
A5
Score Point 1
Annotations
Anchor Paper 5
Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
Correct identification of a as rational and b as irrational (a. is rational . . . b. is
irrational).
The response does not receive credit for correct reasoning to determine rational and
irrational numbers because the explanation of the irrational number is incorrect (it will stop).
The student does not identify the product as an irrational number.
A6
Score Point 1
Annotations
Anchor Paper 6
Score Point 1
This response receives partial credit. The student includes one of the three required
elements:
•
Correct reasoning to determine rational and irrational numbers (rational . . . are able
to be put into a fraction). It is only necessary for the student to provide correct
reasoning for determination of either rational or irrational numbers to receive credit
for this element. The reasoning for both types of numbers is not required.
The student does not identify a as rational or b as irrational. In addition, the product of a
and b is not identified as irrational. The response does show some understanding of the
characteristics of rational numbers
A7
Score Point 0
Annotations
Anchor Paper 7
Score Point 0
This response receives no credit. The student includes none of the three required elements:
The student reverses the identification of a and b (the product of a is irrational . . . product b
is rational). The student does not indicate that the product of the two numbers would be
irrational. The reasoning used for determining whether a number is rational or irrational is
incorrect. Both rational numbers and irrational numbers can be (continued).
A8
Score Point 0
Annotations
Anchor Paper 8
Score Point 0
This response receives no credit. This response does not satisfy any of the three required
elements.
It is not true that both numbers are irrational; therefore, this statement demonstrates a lack
of understanding of differentiating between rational and irrational numbers. There is no
attempt to provide the last two elements of the prompt.
Practice Set
P101 - P105
P101
P102
P103
P104
P105
Practice Set
Paper
Score
P101
3
P102
2
P103
2
P104
1
P105
3