Download (NS) Teacher Key Rational Numbers and Irrational Numbers

Math Grade 8
The Number System (NS)
Teacher Key
Rational Numbers and Irrational Numbers
Benchmarks that are Measured in this Quizlet
Benchmark Code
Benchmark
Item #
8.SMC.NS.1.1-1.a
Show that numbers that are not rational are irrational
1, 2
8.SMC.NS.1.1-2.a
Show that every number has a decimal expansion
3, 4
8.SMC.NS.1.1-3.a
Show that for rational numbers the decimal expansion repeats eventually
5, 6
8.SMC.NS.1.1-4.a
Convert a decimal expansion which repeats eventually into a rational number
7, 14
August 2015
Benchmarks that are Measured in this Quizlet Continued
Benchmark Code
Benchmark
Item #
8.SMC.NS.1.2-1.b
Compare rational approximations of irrational numbers to the size of irrational
numbers
8, 9
8.SMC.NS.1.2-2.b
Locate rational approximations of irrational numbers on a number line diagram
10, 11
8.SMC.NS.1.2-3.b
Estimate the value of expressions by using rational approximations of irrational
numbers
12, 13
8.SMP.6.c
Attend to precision
14
August 2015
MATH GR8 NS QUIZLET 1
Teacher Key
NOTES TO TEACHERS
This series of Quizlets is meant to be used for classroom formative assessment. Different from tools to evaluate student learning
summatively, they are meant to be used by teachers as a part of the instructional process.
Some features of these Quizlets are:
 The Quizlets are fully aligned with the Benchmarks.
 Each Quizlet is has a variety of item types: multiple choice (MC), multiple response (MR), short response (SR) and
performance-based (PB).
 The Quizlets were created using Microsoft® Word so that they can be modified.
 Item stems and graphic organizers can be used to create additional assessments that are aligned to the benchmarks.
Quizlets can be used to identify benchmarks where students are struggling or identify individual students who need additional learning
opportunities.
Item
Teacher Key: Rational Numbers and Irrational Numbers
Benchmark Code
Benchmark
Correct
Answer
Number
of Points
1.
8.SMC.NS.1.1-1.a
Show that numbers that are not rational are irrational
B, E
2
2.
8.SMC.NS.1.1-1.a
Show that numbers that are not rational are irrational
SR
2
3.
8.SMC.NS.1.1-2.a
Show that every number has a decimal expansion
A
1
4.
8.SMC.NS.1.1-2.a
Show that every number has a decimal expansion
SR
4
5.
8.SMC.NS.1.1-3.a
Show that for rational numbers the decimal expansion repeats
eventually
D
1
6.
8.SMC.NS.1.1-3.a
Show that for rational numbers the decimal expansion repeats
eventually
SR
2
7.
8.SMC.NS.1.1-4.a
Convert a decimal expansion which repeats eventually into a rational
number
A
1
8.
8.SMC.NS.1.2-1.b
Compare rational approximations of irrational numbers to the size of
irrational numbers
A
1
9.
8.SMC.NS.1.2-1.b
Compare rational approximations of irrational numbers to the size of
irrational numbers
SR
2
Page 3
MATH GR8 NS QUIZLET 1
Teacher Key
Item
Teacher Key: Rational Numbers and Irrational Numbers Continued
Benchmark Code
Benchmark
Correct
Answer
Number
of Points
10.
8.SMC.NS.1.2-2.b
Locate rational approximations of irrational numbers on a number line
diagram
SR
2
11.
8.SMC.NS.1.2-2.b
Locate rational approximations of irrational numbers on a number line
diagram
SR
2
12.
8.SMC.NS.1.2-3.b
Estimate the value of expressions by using rational approximations of
irrational numbers
D
1
13.
8.SMC.NS.1.2-3.b
Estimate the value of expressions by using rational approximations of
irrational numbers
SR
2
8.SMC.NS.1.1-4.a
Convert a decimal expansion which repeats eventually into a rational
number
PB
8
14.
8.SMP.6.c
Attend to precision
Page 4
MATH GR8 NS QUIZLET 1
Teacher Key
Explained Answers
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 2. Irrational Number (2 points)
Benchmark: 8.SMC.NS.1.1-1.a Show that numbers that are not rational are irrational
Item Stem
Is √3 a rational or irrational number? How do you know?
Responses
√3 is not a perfect square and has an infinite decimal expansion that
does not repeat. Therefore, it is an irrational number.
Points
2 points for correct
response
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 4. Decimal Expansion (4 points)
Benchmark: 8.SMC.NS.1.1-2.a Show that every number has a decimal expansion
Item Stem
What is the 3 digit decimal expansion of the following
numbers?
28
=
4
Responses
Points
28
= 7.000
4
√19 = 4.359
√19 =
2
= 0.666̅
3
2
=
3
4
4
=
11
4
Circle the number(s) that are irrational.
Page 5
4
̅̅̅̅
= 4.363
11
4 points for correct
response
MATH GR8 NS QUIZLET 1
Teacher Key
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 6. Repeating Decimal Expansion (2 points)
Benchmark: 8.SMC.NS.1.1-3.a Show that for rational numbers the decimal expansion repeats eventually
Item Stem
What is the decimal equivalent of
45
111
Responses
Points
2 points for correct
response
̅̅̅̅̅
0. 405
?
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 9. Number Order (2 points)
Benchmark: 8.SMC.NS.1.2-1.b Compare rational approximations of irrational numbers to the size of irrational numbers
Item Stem
List the numbers below in order from least to greatest.
3
,
2
1. 3̅,
√3,
Responses
12
,
11
12
11
1. 3̅,
3
,
2
Points
√3
2 points for correct
response
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 10. Irrational Number on a Number Line (2 points)
Benchmark: 8.SMC.NS.1.2-2.b Locate rational approximations of irrational numbers on a number line diagram
Item Stem
Responses
Place √45 in its approximate location on the number line
below?
Points
2 points for correct
response
Page 6
MATH GR8 NS QUIZLET 1
Teacher Key
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 11. Irrational Number on a Number Line (2 points)
Benchmark: 8.SMC.NS.1.2-2.b Locate rational approximations of irrational numbers on a number line diagram
Item Stem
Responses
Place √20 in its approximate location on the number line
below?
Points
2 points for correct
response
SHORT RESPONSE EXPLAINED ANSWERS
ITEM 13. Distance on a Number Line (2 points)
Benchmark: 8.SMC.NS.1.2-3.b Estimate the value of expressions by using rational approximations of irrational numbers
Item Stem
Responses
√𝑥 – 6.58 = 9.78
√𝑥 = 16.36
(16.36)2
267.65
268
For what integer x, is √𝑥 – 6.53 closest to 9.78 ?
Page 7
Points
2 points for correct
response
MATH GR8 NS QUIZLET 1
Teacher Key
PERFORMANCE-BASED RUBRIC
ITEM 14. Convert Repeating Decimal (8 points)
Benchmark: 8.SMC.NS.1.1-4.a Convert a decimal expansion which repeats eventually into a rational number
Benchmark: 8.SMP.6.c Attend to precision
Item Stem:
Level
Criteria
Example
In addition to the understanding in Level 3.0, the student
4.0
̅?
What fraction is equivalent to 1.205
exhibits in-depth inferences and applications that go
(7-8
̅
x = 1.205
Points)
beyond what was taught.
1000x = 120 + 0. 5̅
y = 0. 5̅
10y = 5 + y
9y = 5
y=
3.0
5
9
(5-6
Points)
1000x = 120 +
1085
9
1085
217
=
9
180
The student exhibits no major errors or omissions at
eighth grade level by:

8.SMC.NS.1.1-4.a Converting repeating decimals to
fractions.

5
9
8.SMP.6.c Following all necessary steps precisely to
determine the equivalent fraction.
1000x =
x=
Your Turn
Choose a different repeating decimal and convert it to a fraction.
2.0
(3-4
Points)
Possible Responses:
(Answers may vary)
̅̅̅
x = 2. ̅15
̅̅̅
x = 2 + 0. ̅15

̅̅̅̅
y = 0. 15
100y = 15 + y
99y = 15
y=
x=
1.0
(1-2
Points)
15
99
x=2+
213
99
The student exhibits major errors or omissions regarding
the more complex ideas and processes, but makes no
major errors or omissions regarding the simpler details
and processes at eighth grade level, including:

8.SMC.NS.1.1-4.a Attempting to convert repeating
decimals to fractions but making errors that do not lead to
a rational number.
15
99
0.0
(0
Points)
Page 8
8.SMP.6.c Following some steps to determine the
equivalent fraction.
With help, the student demonstrates partial
understanding of some of the score 2.0 elements and
some of the score 3.0 elements at the eighth grade level.
Even with help, the student demonstrates little
understanding of skills.