SECURE MATERIAL - Reader Name: _______________________
Tennessee Comprehensive Assessment Program
TCAP/CRA
2014
A2
Phase III
Interior Angle Sum Task
Anchor Set
Copyright © 2014 by the University of Pittsburgh and published under contract with Tennessee State Department of Education by
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Education. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a
database or retrieval system, without the prior written permission of Tennessee Department of Education and the University of Pittsburgh.
Interior Angle Sum Task
The angles inside a polygon formed by pairs of adjacent sides are called interior angles. In the diagram below,
the five interior angles of pentagon ABCDE are shaded.
This table of values shows the sum of the measures of the interior angles of various polygons. For pentagon
ABCDE, m∠A + m∠B + m∠C + m∠D + m∠E = 540 degrees .
a.
Lindsay studies the table and makes the following claim: “Since a five-sided figure has a sum of 540
degrees, the sum of the interior angles of a 10-sided figure must be 2(540) = 1080 degrees.”
Do you agree or disagree with Lindsay’s claim? Explain your reasoning.
b.
Write a recursive rule that can be used to determine the sum of the interior angles of any polygon.
Explain the meaning of each constant and variable in the rule in the context of the problem.
c.
Write an explicit formula that can be used to determine the sum of the interior angles for any polygon.
d.
Use the explicit formula to calculate the sum of the interior angles of a 22-sided figure. Show all work.
© 2014 UNIVERSITY OF PITTSBURGH
3
Scoring Guide
The CCSS for Mathematical Content (2 points)
F-BF.A.1a(x)
Writes the explicit function f(x) = 180(n – 2), or an equivalent form, to
represent the problem situation where n represents the number of
sides of a polygon.
(1 Point)
F-BF.A.1a(z)
Writes the recursive rule an = (an - 1) + 180, or an equivalent form, to
represent the problem situation.
(1 Point)
_____
The CCSS for Mathematical Practice (4 points)
MP1 Makes sense of the problem by extending the table, looking for patterns in the
table, drawing diagrams, and/or using the formula to calculate the sum of the
interior angles.
(1 Point)
(MP1: Make sense of problems and persevere in solving them.)
MP2 Explains correctly the meaning of each term and constant value in the recursive
rule an = (an - 1) + 180, or its equivalent form. In this case the constant 180
represents the increase in degrees from one polygon to the next, or the number of
degrees in a triangle. The term an - 1 represents the sum of the interior angles of a
polygon with one less side. The student may explain the meaning of the terms in
the explicit formula if the explicit formula is given in Part B.
(1 Point)
(MP2: Reason abstractly and quantitatively)
_____
MP3 Argues that Lindsay is incorrect for a reason such as the following:
- explaining that the function is not a direct variation and therefore the
relationship between the values is not multiplicative;
- extending the table and determining that the sum of the interior angles of a
10-sided figure is not 1080 degrees; or
- using the explicit or recursive formula to calculate the sum of the interior
angles of a 10-sided figure and noting that it is not 1080 degrees.
May have minor calculation errors.
(1 Point)
(MP3: Construct viable arguments and critique the reasoning of others)
MP6 Uses the explicit function correctly to determine the sum of the interior angles of a
22-sided figure; uses precise notation when writing the explicit and recursive rules.
(1 Point)
(MP6: Attend to precision)
_____
TOTAL POINTS: 6
© 2014 University of Pittsburgh
The CCSS for Mathematical Content Addressed In This Task
Building Functions
F-BF
Build a function that models a relationship between two quantities
F-BF.A.1a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and
specific modeling standards appear throughout the high school standards indicated with a star. Where an
entire domain is marked with a star, each standard in that domain is a modeling standard.
★
The CCSS for Mathematical Practice*
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
* Gray type indicates Mathematical Practices not addressed in this assessment.
© 2014 University of Pittsburgh
A-1a
Litho#: 00900060018
A-1b
Litho#: 00900060018
A-1c
Litho#: 00900060018
Anchor 1
Litho 00900060018
Total Content Points: 2
(F-BF.A.1a(x), F-BF.A.1a(z))
Total Practice Points: 4
(MP1, MP2, MP3, MP6)
In Part C, the student writes an explicit formula (F(x) = (x – 2)180) that can be used to find the
sum of the interior angles for any polygon (F-BF.A.1a(x)). In Part B, the student writes a
recursive rule (Tn = Tn – 1 + 180) that represents the problem situation (F-BF.A.1a(z)). The
student recognizes the pattern in the table in Part A, and continues the pattern for each number of
angles up to 10 (MP1). In Part B, the student identifies the meaning of each term, using a key
and explanations with arrows to explain the meaning (MP2). In Part A, the student identifies a
flaw with Lindsay’s reasoning (“Lindsay doubled the total sum from 5, even though there was no
sum for 1 or 2 angles”) to explain why Lindsay is incorrect (MP3). The student correctly finds
the answer of 3600 in Part D using the explicit function, and uses correct notation throughout the
response (MP6).
Total Awarded Points: 6 out of 6
A-2a
Litho#: 01100060018
A-2b
Litho#: 01100060018
A-2c
Litho#: 01100060018
Anchor 2
Litho 01100060018
Total Content Points: 2
(F-BF.A.1a(x), F-BF.A.1a(z))
Total Practice Points: 3
(MP1, MP2, MP3)
In Part C, the student writes an explicit formula (f(x) = (x – 2)180) that can be used to find the
sum of the interior angles for any polygon (F-BF.A.1a(x)). In Part B, the student writes a
recursive rule that can be used to find the sum of a polygon with x sides (Tx = Tx – 1 + 180)
(F-BF.A.1a(z)). The student continues the table in Part A to find the sum of the angles in a
10-sided polygon (MP1). In Part B, the student explains the terms of the recursive rule (“Total
Sum of x angles = Total sum of x – 1 angles + 180 . . . It goes up 180 when you add a new
angle”) (MP2). The student disagrees with Lindsay in Part A, and by showing the number of
angles that would result in a sum of 1080 degrees, and the sum of 10 interior angles, indicates
that Lindsay is incorrect (MP3). The student makes an error when calculating the sum of the
interior angles of a 22-sided figure in Part D (no credit for MP6).
Total Awarded Points: 5 out of 6
A-3a
Litho#: 01000060018
A-3b
Litho#: 01000060018
A-3c
Litho#: 01000060018
Anchor 3
Litho 01000060018
Total Content Points: 2
(F-BF.A.1a(x), F-BF.A.1a(z))
Total Practice Points: 3
(MP1, MP3, MP6)
In Part C, the student writes an explicit formula (y = (x – 2)180) that can be used to find the sum
of the interior angles for any polygon (F-BF.A.1a(x)). In Part B, the student writes a recursive
rule (an = an – 1 + 180) that can be used to find the sum of the interior angles for any polygon to
describe the situation (F-BF.A.1a(z)). The student uses the pattern apparent in the table in Part A
to find the sum of the interior angles in a figure with 10 sides (MP1). In Part B, the student
makes a general statement about the increase in the sum of the interior angles, but does not
explain the meaning of the terms in the recursive rule or in the explicit formula from Part C
(no credit for MP2). The student disagrees with Lindsay in Part A, indicating that Lindsay is
incorrect because the sum of the interior angles of a 10-sided figure is 1440, not 1080, and shows
work indicating how to find the total sum (MP3). The student uses correct notation for the
recursive rule in Part B, writes a correct equation in Part C, and uses the formula to find the
correct sum in Part D (MP6).
Total Awarded Points: 5 out of 6
A-4a
Litho#: 01200060018
A-4b
Litho#: 01200060018
A-4c
Litho#: 01200060018
Anchor 4
Litho 01200060018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 3
(MP1, MP3, MP6)
In Part C, the student writes the expression for an explicit formula ((x – 2)180) that can be used
to find the sum of the interior angles for any polygon (F-BF.A.1a(x)). In Part B, the student
makes a correct general statement but does not state a recursive rule (no credit for F-BF.A.1a(z)).
The student continues the pattern shown in the table in Part A to find the sum of the interior
angles of a figure with 10 sides (MP1). The student does not explain the terms of a recursive rule
in Part B, or the terms of the explicit formula in Part C (no credit for MP2). In Part A, the student
disagrees with Lindsay, and demonstrates the correct reasoning to find the correct sum of the
interior angles of a 10-sided figure (MP3). The student uses the explicit formula to find the
correct sum in Part D, uses precise language to explain the concept behind the increase of the
sums as the number of interior angles increases in Part B, and writes a correct expression in
Part C (MP6).
Total Awarded Points: 4 out of 6
A-5a
A-5b
A-5c
Anchor 5
Litho 000200690018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 3
(MP1, MP3, MP6)
In Part C, the student writes an explicit formula ((n – 2)(180) = s) that can be used to find the
sum of the interior angles for any polygon (F-BF.A.1a(x)). The student does not write a recursive
rule to represent the situation in Part B (no credit for F-BF.A.1a(z)). In Part A, the student
derives the formula describing the sum of the interior angles of a polygon (MP1). The student
does not clearly explain the terms of a recursive rule determining the sum of the interior angles
of a polygon, or the terms of the explicit formula presented in Part B. The student attempts an
explanation in Part B, but does not explain the meaning of the constant 180 in this context
(no credit for MP2). The student disagrees with Lindsay in Part A, demonstrating that the total
Lindsay found for the sum of the interior angles of a figure with 10 sides is incorrect (MP3). The
student correctly uses the formula to find the sum of the interior angles of a figure with 22 sides
in Part D, explains with precision how to find the sum on the interior angles of a polygon in
Part B, and uses precise notation when writing an equation in Part C (MP6).
Total Awarded Points: 4 out of 6
A-6a
Litho#: 02400060018
A-6b
Litho#: 02400060018
A-6c
Litho#: 02400060018
Anchor 6
Litho 02400060018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 2
(MP1, MP3)
In Part C, the student writes an explicit formula (Sum = (n – 2)180) that can be used to find the
sum of the interior angles for any polygon (F-BF.A.1a(x)). The student does not write a recursive
rule to represent the situation in Part B (no credit for F-BF.A.1a(z)). In Part A, the student
continues the pattern shown in the table to find that the sum of the interior angles of a figure with
8 sides is 1080 (MP1). The student does not explain the terms of a recursive rule describing the
sum of the interior angles of a polygon (no credit for MP2). In Part A, the student disagrees with
Lindsay and explains that the function is not a direct variation, so the relationship between the
values is not multiplicative (MP3). The student makes an error when calculating the sum of the
interior angles of a 22-sided figure in Part D (no credit for MP6).
Total Awarded Points: 3 out of 6
A-7a
A-7b
A-7c
Anchor 7
Litho 00050060018
Total Content Points: 0
Total Practice Points: 2
(MP1, MP3)
In Part C, the student does not write an explicit formula that can be used to find the sum of the
interior angles for any polygon (no credit for F-BF.A.1a(x)). The student does not write a
recursive rule to represent the situation in Part B (no credit for F-BF.A.1a(z)). In Part A, the
student looks for and recognizes a pattern in the table that indicates Lindsay is incorrect (“If her
idea was right then a 6 sided polygon woud be 2(180) OR 360 degrees And that is the sum of a
4 sided polygon on the chart”) (MP1). The student does not explain the terms of a recursive rule
describing the sum of the interior angles of a polygon (no credit for MP2). In Part A, the student
disagrees with Lindsay and uses the pattern in the table to support that disagreement (MP3). The
student does not write or explain a recursive rule in Part B, does not find the sum of the interior
angles of a figure with 22 sides in Part D, and does not write an explicit formula for finding the
sum of the interior angles of any polygon in Part C (no credit for MP6).
Total Awarded Points: 2 out of 6
A-8a
Litho#: 01700060018
A-8b
Litho#: 01700060018
A-8c
Litho#: 01700060018
Anchor 8
Litho 01700060018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 1
(MP1)
In Part C, the student writes an explicit formula ((x – 2) × 180) that can be used to find the sum
of the interior angles for any polygon (F-BF.A.1a(x)). The student does not write a recursive rule
to represent the problem situation in Part B (no credit for F-BF.A.1a(z)). In Part A, the student
recognizes the pattern in the table and uses it to write a correct formula for finding the sum of the
interior angles of a figure; even though an error in calculation leads to the student finding the
incorrect sum for a figure with 10 sides, the logic is correct (MP1). The student does not explain
the meaning of the terms of a recursive rule or of the explicit formula representing the situation
(no credit for MP2). In Part A, the student incorrectly agrees with Lindsay due to an error in
calculation (no credit for MP3). The student does not find the sum of the interior angles of a
figure with 22 sides in Part D, does not write a recursive rule in Part B, and does not present any
explicit explanations to demonstrate precision (no credit for MP6).
Total Awarded Points: 2 out of 6
A-9a
A-9b
A-9c
Anchor 9
Litho 00090060018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 1
(MP1)
In Part C, the student writes the equation “SUM = (A1 – 3) × 180 + 180” that can be used to find
the sum of the interior angles for any polygon (F-BF.A.1a(x)). The student does not write a
recursive rule to model the situation in Part B (no credit for F-BF.A.1a(z)). In Part A, the student
continues the table by adding 180 to the sum for each successive number of sides (MP1). The
student does not explain the terms of a recursive rule or of the explicit formula for finding the
sum of the interior angles of a polygon (no credit for MP2). Although the student continues the
table to show a 10-sided figure in Part A, there is no explicit agreement or disagreement with
Lindsay (no credit for MP3). In Part D, the student correctly uses the formula to solve for the
sum of the interior angles of a figure with 22 sides, but in Parts B and C the student uses
confusing notation in writing the formulas (no credit for MP6).
Total Awarded Points: 2 out of 6
A-10a
Litho#: 01300060018
A-10b
Litho#: 01300060018
A-10c
Litho#: 01300060018
Anchor 10
Litho 01300060018
Total Content Points: 1
(F-BF.A.1a(x))
Total Practice Points: 0
In Part C, the student writes an explicit formula (y = (180(x – 2)) that can be used to find the sum
of the interior angles for any polygon (F-BF.A.1a(x)). The student does not write a recursive rule
to represent the situation in Part B (no credit for F-BF.A.1a(z)). The student does not show any
work or explanation in Part A indicating how to find the sum of the interior angles (no credit
for MP1). The student does not explain the terms of a recursive rule or of an explicit formula
(no credit for MP2). In Part A, the student disagrees with Lindsay but does not give a sufficient
explanation why Lindsay is incorrect (no credit for MP3). In Part D, the student does not use the
explicit formula to find the sum of the interior angles of a figure with 22 sides, and the
explanation in Part B lacks precision (no credit for MP6).
Total Awarded Points:
1 out of 6