Download High School Claim 3 Specifications 11 Version 2.0

High School Claim 3 Specifications
Task Model 1
DOK Level 2
Test
propositions
or conjectures
with specific
examples
Target A
Task Expectations:
The student is asked to give an example that refutes a proposition or conjecture; or
The student is asked to give an example that supports a proposition or conjecture.
[Note: Use appropriate mathematical language in asking students for a single example. While a single
example can be used to refute a conjecture, it cannot be used to prove one is always true.]
Example Item 1:
Primary Target 3A (Content Domain F-IF), Secondary Target 1X (CCSS F-IF.2), Tertiary Target 3F
A baseball player hits a ball toward a 20-foot wall 400 feet
away. The path of the ball is modeled by the function,
( )
,
where x is the distance of the ball, in feet.
Claire claims the domain of the function is the set of
all real numbers.
Steve claims the domain of the function is the set of all
positive numbers.
Part A: Plot a point that demonstrates that Claire is
incorrect. Select None if there is not point that
demonstrates that Claire is incorrect.
Part B: Plot a point that demonstrates that Steve is incorrect.
Select None if there is not a point that demonstrates that
Steve is incorrect.
Part C: Select the person with the correct domain or choose Neither.
Rubric: (2 points) The student is able to identify a point on both graphs that demonstrates Claire and Steve
are both incorrect.
(1 point) The student is able to interpret one of them as incorrect, but not both, and can place a point that
demonstrates where the student’s thinking is flawed.
Response Type: Graphing and Hot Spot
11
Version 2.0
High School Claim 3 Specifications
Task Model 1
Example Item 2:
Primary Target 3A (Content Domain G-SRT), Secondary Target 1X (CCSS G-SRT.2), Tertiary Target 3F
DOK Level 2
Test
propositions
or conjectures
with specific
examples
The radius of sphere Y is twice the radius of
sphere X. A student claims that the volume
of sphere Y must be exactly twice the volume
of sphere X.
Part A: Drag numbers into the boxes to create
one example to evaluate the student’s claim.
Target A
Part B: Decide whether the student’s claim is
True, False, or whether it Cannot be determined.
Select the correct option.
Rubric: (1 point) The student supplies an example of two radii and missing numbers for the volumes that
makes the conjecture false (e.g., radii are 1 and 2, missing numbers are 1 and 8; radii are 2 and 4, missing
numbers are 8 and 64; radii are 1 and 3, missing numbers are 1 and 27; etc.) and responds with “False” in
part B.
Response Type: Drag and Drop
12
Version 2.0
High School Claim 3 Specifications
Task Model 2
DOK Levels 3,
4
Task Expectations: The student is presented with a mathematical phenomenon and a conjecture. Student
is asked to identify or construct reasoning that justifies or refutes a conjecture.
Example Item 1:
Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.6), Tertiary Target 3C
Construct,
autonomously,
chains of
reasoning that
will justify or
refute
conjectures
Target B
Jose and Tina are writing a program for a computer game.
They need to move Triangle A to Triangle A’.
To move Triangle A to Triangle A’,
Jose thinks:
a sequence of three transformations
must be performed, and
there is only one possible way to do it.
Tina thinks:
there are other sequences of transformations
that will work, and
it can be done using fewer than three
transformations.
Part A:
Describe a sequence of three transformations that maps
Triangle A onto Triangle A’ to support Jose’s thinking.
Part B:
If possible, support Tina’s thinking by describing a sequence
of fewer than three transformations that maps Triangle A onto Triangle A'.
Enter your response to Part A and Part B into the response box. Be sure to number the transformations in
the order they should be performed (e.g., 1, 2, 3). Label each part of your response with “Part A” and “Part
B.”
Rubric: (2 points) The student is able to generate three correct translations to support Jose’s thinking and
one or two transformations to support Tina’s thinking.
(1 point) The student is able to generate correct transformation(s) to support either Jose’s thinking or Tina’s,
but not both.
13
Version 2.0
High School Claim 3 Specifications
Task Model 2
DOK Levels 3,
4
Construct,
autonomously,
chains of
reasoning that
will justify or
refute
conjectures
Exemplar5:
Part A
1. reflection across the x-axis
2. reflection across the y-axis
3. translation 2 units to the right
Part B
1. rotation 180° about the origin
2. translation 2 units to the right
OR
1. rotation 180° about the point (1, 0)
OR Any response for parts A and B that results in a correct mapping of Triangle A to Triangle A’ that uses
the given constraints.
Response Type: Short Text (hand scored)
Target B
5
Exemplars only represent possible solutions. Typically, many other solutions/responses may receive full credit. The full range of acceptable responses is
determined during rangefinding and/or scoring validation.
14
Version 2.0
High School Claim 3 Specifications
Task Model 2
DOK Levels 3,
4
Construct,
autonomously,
chains of
reasoning that
will justify or
refute
conjectures
Example Item 2:
Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.C), Tertiary Target 3F
(Source: Adapted from “The Envelope”, Smarter Mathematics Content Specifications, p. 70)
A folded envelope is shown in the diagram
as rectangle PQRS. Segments PR and SQ
are diagonals of the envelope.
Prove that when the rectangular envelope
(PQRS) is unfolded, the shape obtained (ABCD)
is a rhombus.
Target B
Rubric: (4 points) The student is able to justify each step in the proof, making sure to provide adequate
reasoning to support the claim that all four sides of ABCD are congruent and opposite sides are parallel.
Exemplar: A rhombus is a parallelogram with four congruent sides. As shown in the diagram, points A, B, C,
and D all represent the same point where the diagonals of PQRS bisect each other. I’ll name this point “T” to
make the connection to points A, B, C, and D in the unfolded envelope. I know angles QTR and RTS are
supplementary, because they form a straight line. Angles RTS and STP are supplementary as well, because
they also form a straight line. Since point T is the same point as C and D, it follows that angles QCR and RDS
supplementary, which means side BC is parallel to side AD. Also, since point T is the same point as A, it
follows that and RDS and SAP are supplementary, which means side AB is parallel to side DC. This makes
quadrilateral ABCD a parallelogram, because opposite sides BC || AD and AB || DC.
Now we need to prove one pair of consecutive sides is congruent. Using the property that diagonals of a
rectangle are congruent and bisect each other, PT is congruent to ST. PA and PB are both equal to PT. SA
and SD are both equal to ST. PT = ST, then PA + PB = AB and SA + SD = AD or 2PT = AB and 2ST = AD.
By substitution, AB = AD and the sides are also congruent. Given two adjacent sides of the parallelogram
ABCD are congruent, and their opposite sides must also be congruent, proves that ABCD is a rhombus.
This exemplar response represents only one possible solution. Other correct responses are possible.
Response Type: Short Text (hand scored)
15
Version 2.0
High School Claim 3 Specifications
Task Model 2
Example Item 3:
Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.11), Tertiary Target 3F
DOK Levels 3,
4
Given: ̅̅̅̅
̅̅̅̅ and ̅̅̅̅
̅̅̅̅
Statements
Construct,
autonomously,
chains of
reasoning that
will justify or
refute
conjectures
Prove: If a quadrilateral has a pair of sides that are
both congruent and parallel, then the quadrilateral
is a parallelogram.
Target B
Complete the proof by providing reasons that justify
each statement.
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
Reasons
Given
Given
ABCD is a parallelogram
Rubric:
(3 points) The student correctly completes the proof with reasons to support each statement.
(2 points) The student is able to correctly complete four of the reasons for each statement.
(1 point) The student is able to correctly complete three of the reasons for each statement.
Exemplar:
Alternative interior angles are congruent
Reflexive property
Side-Angle-Side (SAS)
Corresponding parts of congruent triangles are congruent
Both pairs of opposite sides are congruent; definition of a parallelogram
Note: This exemplar response represents only one possible solution. Other correct responses are possible.
Response Type: Fill-in Table (hand scored)
16
Version 2.0
High School Claim 3 Specifications
Task Model 3
DOK Levels 2,
3
State logical
assumptions
being used
Target C
Task Expectations: The student is asked to connect a logical basis to its resulting conjecture. The student
is presented with a mathematical or real-world scenario where an assumption is made in order to find the
solution.
Example Item:
Primary Target 3D (Content Domain N-RN), Secondary Target 1B (CCSS N-RN.3), Tertiary Target 3F
Drag values for and into the boxes to
make the paired statements true for both
and
If none of the values make both
statements true, leave the boxes empty for
that pair of statements.
Statements
is an irrational number
is a rational number
√
is a rational number
√
√
√
Rubric: (1 point) The student drags correct
combinations of values that result in the indicated
rational and irrational perimeters and areas of the
rectangle. (e.g., see one possible solution at right).
Other correct responses are possible.
is an irrational number
is an irrational number
is an irrational number
Statements
Example
is an irrational number
√
is a rational number
is a rational number
is an irrational number
Response Type: Drag and Drop
is an irrational number
is an irrational number
17
Example
√
√
√
√
√
Version 2.0
High School Claim 3 Specifications
Task Model 4
DOK Levels 2,
3
Use the
technique of
breaking an
argument into
cases
Target D
Task Expectations: The student is given different cases to consider and the student uses or shows
reasoning to support the cases.
Example Item:
Primary Target 3D (Content Domain F-IF), Secondary Target 1L (CCSS F-IF.B), Tertiary Target 3F
A student examines two graphs representing functions
( ) and
( ). The student notices that the
graphs of the functions ( )
and ( )
both have a -intercept at the point (0, 5). The student
makes the following claim:
“For any constant , the location of the -intercepts for the graphs of
( ) and
( ) is the same point.”
Show that this claim is true by giving the -value, in terms of , of the ordered pair (0, ?) that represents
the -intercept for the graphs of
( ) and
( ).
Rubric: (1 point) The student is able to identify the correct -intercept value (e.g., 5 ).
Response Type: Equation/Numeric
18
Version 2.0
High School Claim 3 Specifications
Task Model 5
DOK Levels 2,
3
Distinguish
correct
reasoning
from flawed
reasoning
Task Expectations:
The student is presented with valid or invalid reasoning. If the reasoning is flawed, the student will
explain or correct the flaw.
The student is asked to select the condition(s) for which an argument does or does not always apply.
Two or more approaches or chains of reasoning are given and the student is asked to identify the correct
method and justification OR identify the incorrect method/reasoning and the justification.
Example Item:
Primary Target 3E (Content Domain N-Q), Secondary Target 1C (CCSS N-Q.A), Tertiary Target Grade 7 1A
Sherry wants to enlarge a photograph to 300%
Target E
of its original size.
The machine she is using can only make
enlargements for the following percentages:
100%, 125%, 150%, and 200%.
Sherry thinks that she should enlarge the photograph
by 100% and then by 200% to get a total enlargement
of 300%.
%
Sequence of Enlargements
1:
%
2:
%
3:
%
4:
%
5:
%
Decide if Sherry is correct. If Sherry is correct, drag the 100 and 200 from the palette into the first two
answer spaces.
If Sherry is incorrect, drag a sequence of enlargements she can use to get a total enlargement of 300%.
Each enlargement percentage may be used more than once. If a sequence is not needed, leave the box
blank.
Rubric: The student drags the percentages into the appropriate boxes to show Sherry is incorrect (e.g.,
200, 150). Other correct responses are possible.
Response Type: Drag and Drop
19
Version 2.0
High School Claim 3 Specifications
Task Model 6
Task Expectations: The student uses concrete referents to help justify or refute an argument.
DOK Levels 2,
3
Example Item:
Base
arguments on
concrete
referents such
as objects,
drawings,
diagrams, and
actions
Target F
Primary Target 3E (Content Domain N-RN), Secondary Target 1B (CCSS N-RN.3), Tertiary Target 3C
Ashley claims that when you multiply two different square roots together, the product is always rational.
For example, √ √ =√
= 6 and √ √ = √ = 9.
She also claims that that when you multiply two different cube roots together, the product is always
irrational. For example, √ √ = √ ≈ 3.3019 and √ √ = √ ≈ 4.3027
Which statement correctly classifies Ashley’s claims and provides appropriate reasoning?
A. Ashley is correct because her examples support both claims.
B. Ashley is correct about the product of square roots always being rational, but the product of cube roots
can sometimes be rational.
C. Ashley is incorrect about the product of square roots always being rational, but she is correct that the
product of cube roots is always irrational.
D. Ashley is incorrect because sometimes the product of square roots can be irrational, and sometimes the
product of cube roots can be rational.
Rubric: (1 point) The student identities the correct statement (e.g., D).
Response Type: Multiple Choice, single correct response
20
Version 2.0
High School Claim 3 Specifications
Task Model 7
Task Expectations: The student is asked to construct the conditions for which an argument does and does
not apply.
DOK Level 3, 4
Example Item 1:
Determine
conditions
under which
an argument
does and does
not apply
Target G
Primary Target 3G (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.9), Tertiary Target 3C,
Quaternary Target 3A
A geometry student made this claim:
If two lines are cut by a transversal,
then alternate interior angles are
congruent.
Part A:
Draw a diagram that shows congruent
alternate interior angles or select None
if there is not a situation to support the
student’s claim.
Part B:
Draw a diagram that shows alternate interior
angles that are not congruent or select None
if there is not a situation to support the
student’s claim.
Rubric: (1 point) The student draws a transversal through two parallel lines to create congruent alternate
interior angles in Part A and through two non-parallel lines for Part B.
Response Type: Graphing and Hot Spot
21
Version 2.0
High School Claim 3 Specifications
Task Model 7
Example Item 2:
Primary Target 3G (Content Domain N-RN), Secondary Target 1X (CCSS N-RN.1), Tertiary Target 3A
DOK Level 3, 4
Determine
conditions
under which
an argument
does and does
not apply
Target G
An inequality is shown below.
√
Part A:
Determine the positive values of m for which the inequality is true.
Enter your response as an inequality in the first response box.
Part B:
Determine the positive values of m for which the inequality is false.
Enter your response as an inequality in the second response box.
Rubric: (2 points) The student provides the values that make the given inequality true (e.g.,
) and
false (e.g.,
).
(1 point) The student errors in the use of the inequality signs (uses > instead of or instead of <) but
otherwise has the correct range of values. OR The student is able to provide an example of where the
statement is true (e.g., a positive value greater than or equal to 1) and a number between 0 and 1 that
makes the statement false.
Response Type: Equation/Numeric
22
Version 2.0
High School Claim 3 Specifications
Task Model 7
Task Expectations: The student must determine whether a proposition or conjecture is True for all, True
for some, or Not true for any and/or provide justification to support their conclusions.
DOK Level 3, 4
Example Item 1:
Primary Target 3G (Content Domain A-REI), Secondary Target 1X (CCSS A-REI.A), Tertiary Target 3A
Determine
conditions
under which
an argument
does and does
not apply
Determine whether each statement is
true for all values of x,
true for some values of x, or
not true for any value of x
True for
all
Statement
Target G
If
If
If (
True for
some
Not true
for any
= 9, then √ =
, then √ =
) = , then √ = (
)
Rubric: (1 point) The student correctly selects True for all, True for some, or Not true for any for each
statement correctly (e.g., see below).
Statement
If
If
If (
True for
all
True for
some
Not true
for any
= 9, then √ =
, then √ =
) = , then √ = (- )
Response Type: Matching Tables
23
Version 2.0
High School Claim 3 Specifications
Task Model 7
Example Item 2:
Primary Target 3G (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.10), Tertiary Target 3A
DOK Level 3, 4
Consider all triangles that satisfy the given conditions:
Determine
conditions
under which
an argument
does and does
not apply
D is a point on the side of △ABC
̅̅̅̅ ̅̅̅̅
=
The picture shows one triangle that satisfies these conditions.
Target G
Determine whether each statement is
true for all triangles that satisfy the given conditions
true for some triangles that satisfy the given conditions
not true for any triangle that satisfies the given conditions
Statement
True for
all
True for
some
Not true
for any
̅̅̅̅ is perpendicular to ̅̅̅̅ .
m ABD is less than m BAD
= ½BC
m ACB = ½(m ABD)
Rubric: (1 point) Student correctly selects True for all, True for some, or Not true for any for each
statement of equation.
24
Version 2.0
High School Claim 3 Specifications
Statement
True for
all
True for some
Not true for
any
̅̅̅̅ is perpendicular to ̅̅̅̅ .
m ABD is less than m BAD
= ½BC
m ACB = ½(m ABD)
Response Type: Matching Tables
25
Version 2.0