Sample tasks from:
Algebra Assessments
Through the Common Core
(Grades 6-12)
A resource from
The Charles A Dana Center at
The University of Texas at Austin
2011
About the Dana Center Assessments
More than a decade ago, the Charles A. Dana Center began work on collections of assessment
tasks that could be used by teachers at many grade levels to enable them to assess student
learning continually as they enacted mathematics instruction. These assessments, developed in
collaboration with mathematics educators, are designed to make clear to teachers, students, and
parents what is being taught and learned about the most central mathematical concepts at each
grade or in each course. Published by the Dana Center as a series of books (available for order
here: http://www.utdanacenter.org/products/math.php), these collections of assessments
eventually encompassed tasks for middle school, Algebra I, Geometry, and Algebra II.
These tasks have been used and tested by tens of thousands of educators and their students. Now
the Dana Center has published a selection of these tasks in new editions: 118 tasks in Algebra
Assessments Through the Common Core (Grades 6-12) and 57 tasks in Geometry Assessments
Through the Common Core (Grades 6-12).
The alignment to the Common Core State Standards for Mathematics of 164 existing Dana Center
mathematics assessment tasks, and the development of 11 new tasks aligned to the standards, was
made possible by a grant from Carnegie Corporation of New York and from the Bill and Melinda
Gates Foundation. (Gates Foundation grant number OPP-48458 and Carnegie Corporation of
New York grant number B 8312.) The statements made and views expressed are solely the
responsibility of the authors.
We thank the Gates Foundation and Carnegie Corporation of New York for their support of
Improving Mathematics and Literacy Instruction at Scale, a collaboration between the Aspen
Institute (literacy) and the Dana Center (mathematics). We used funds from this grant to align
mathematics assessments to the "Standards for Mathematical Content" and "Standards for
Mathematical Practice" of the Common Core State Standards for Mathematics and to publish
these assessments to this UDLN website. We thank these funders for their generous support of
this work.
To support the adoption of the Common Core State Standards for Mathematics, the Dana Center
is pleased to make available here two sample tasks from Algebra Assessments Through the
Common Core (Grades 6-12): “Mosaics” and “Paintings on a Wall.”
About Mosaics
Standard for Mathematical Practice #4 calls for students to model with mathematics. Students
should be using the mathematics they know to solve problems that arise in everyday life. This
task presents a situation in which tiles are used to create mosaics in a growing pattern. Students
are asked to represent the relationship between the mosaic number and the number of tiles in the
mosaic using multiple mathematical representations, making connections between the situation
and the mathematical representations clear. Students use their models to answer questions and
make predictions about the situation, and adjust their models based on changes in the situation.
About Paintings on a Wall
Standard for Mathematical Practice #5 calls for students to use appropriate tools strategically.
Students should consider different tools available and decide when those tools might be helpful in
solving a problem. In this task, students solve a problem using algebraic methods and technology,
and then explain their solutions. Students can choose which representation to use in their
explanations and therefore must consider which tools will be most useful in their explanations. Chapter 2:
Function Fundamentals
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Algebra Assessments Through
the Common Core (grades 6-12), 2011
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Algebra Assessments Through
the Common Core (grades 6-12), 2011
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Algebra Assessments Through
the Common Core (grades 6-12), 2011
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at The University of Texas at Austin
-\UJ[PVU-\UKHTLU[HSZ
Ch 2–13
Algebra Assessments Through
the Common Core (grades 6-12), 2011
4VZHPJZ
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at The University of Texas at Austin
Ch 2–14
Algebra Assessments Through
the Common Core (grades 6-12), 2011
-\UJ[PVU-\UKHTLU[HSZ
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Chapter 2:
Function Fundamentals
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The Charles A. Dana Center
at The University of Texas at Austin
-\UJ[PVU-\UKHTLU[HSZ
Ch 2–15
Algebra Assessments Through
the Common Core (grades 6-12), 2011
4VZHPJZ
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Function Fundamentals
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The Charles A. Dana Center
at The University of Texas at Austin
Ch 2–16
Algebra Assessments Through
the Common Core (grades 6-12), 2011
-\UJ[PVU-\UKHTLU[HSZ
4VZHPJZ
;LHJOLY5V[LZ
Chapter 2:
Function Fundamentals
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The Charles A. Dana Center
at The University of Texas at Austin
-\UJ[PVU-\UKHTLU[HSZ
Ch 2–17
Algebra Assessments Through
the Common Core (grades 6-12), 2011
4VZHPJZ
;LHJOLY5V[LZ
Chapter 2:
Function Fundamentals
35
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Rule A, y = x + 3
Rule B, y = 4x + 3
Rule C, y = 4x + 6
The Charles A. Dana Center
at The University of Texas at Austin
Ch 2–18
Algebra Assessments Through
the Common Core (grades 6-12), 2011
-\UJ[PVU-\UKHTLU[HSZ
Chapter 8:
Rational Functions
Paintings on a Wall
In order to optimize the viewing space for its patrons, a museum has placed
size restrictions on rectangular paintings that will be hung on a particular wall.
The perimeter of a painting must be between 64 inches and 100 inches,
inclusive. The area of the painting must be between 200 square inches and
500 square inches.
1. You are in charge of determining possible perimeter and area combinations
for paintings to be hung on the wall. Write inequalities to describe the
perimeter and area restrictions in terms of the length and the width of the
rectangles. Graph the resulting system.
2. Algebraically and with technology, determine the vertices of the region
GH¿QHG by the LQHTXDOLWLHV LQ SUREOHP 1.
3. Describe the location of the points on your graph where the dimensions of
the painting result in each of the following:
a. The perimeter and area are acceptable.
b. The perimeter is too short or too long, but the area is acceptable.
c. The perimeter is acceptable, but the area is too small or too large.
d. Neither the perimeter nor the area is acceptable.
Explain how you arrived at your responses.
The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Ch 8–9
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
5V[LZ
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Scaffolding Questions:
(A-CED) Create equations
that describe numbers or
relationships
‡ :KDWLQIRUPDWLRQisknown?
3. Represent constraints by
equations or inequalities,
and by systems of equations
and/or inequalities, and
interpret solutions as viable
or non-viable options in
a modeling context. For
example, represent inequalities
describing nutritional and cost
constraints on combinations of
different foods.*
‡ :KDWFRPSRXQGLQHTXDOLW\FDQyouZULWHWRGHVFULEH
WKHUHVWULFWLRQVonWKHSHULPHWHU"
(A-REI) Solve equations and
inequalities in one variable
‡ :KDWIXQFWLRQVPXVWEHJUDSKHGWRPRGHODFRPELQHG
inequality such as 2 b x y b 5 ?
4. Solve quadratic equations in
one variable.
b. Solve quadratic equations by
inspection (e.g., for x2 = 49),
WDNLQJVTXDUHURRWVFRPSOHWLQJ
the square, the quadratic
formula and factoring, as
appropriate to the initial form of
the equation. Recognize when
the quadratic formula gives
complex solutions and write
them as a ± bi for real numbers
a and b.
(A-REI) Represent and solve
equations and inequalities
graphically
11. Explain why the
x-coordinates of the points
where the graphs of the
equations y = f(x) and y = g(x)
intersect are the solutions of
WKHHTXDWLRQI[=J[¿QG
the solutions approximately,
The Charles A. Dana Center
at The University of Texas at Austin
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that describe the boundaries of the region containing
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the boundary functions and describe whether the
perimeter and area meet the requirements in that
region.
Sample Solutions:
1. Let l = the length in inches of a painting and w = the width
in inches of the painting.
Since the perimeter of the painting must be between 64
LQFKHVDQG100LQFKHVZHknowWKDW
64 b 2(l w ) b 100
which gives
32 b l w b 50
32 w b l b 50 w
Ch 8–10
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
Since the area of the painting must be between 200
square inches and 500 square inches, we have
200 b lw b 500
which gives
200
500
bl b
w
w
On the graphing calculator, the length will be represented
by Y and the width will be represented by x.
Function Rule:
Calculator Rule:
l 32 w
l 50 w
200
l
w
500
l
w
Y1 32 x
Y2 50 x
200
Y3 x
500
Y4 x
Here is the calculator graph of the system with window
settings 0 b x b 47, - 10 b y b 50:
The region that is the solution set to the system of
inequalities is the closed region between the two lines
and the two curves.
e.g., using technology to graph
the functions, make tables
of values, or ¿QG successive
approximations. Include cases
where f(x) and/or g(x) are
linear, polynomial, rational,
absolute value, exponential,
and logarithmic functions.*
12. Graph the solutions to
a linear inequality in two
variables as a half-plane
(excluding the boundary in
the case of a strict inequality),
and graph the solution
set to a system of linear
inequalities in two variables
as the intersection of the
corresponding half-planes.
(F-IF) Interpret functions that
arise in applications in terms
of the context
4. For a function that models
a relationship between two
quantities, interpret key
features of graphs and tables
in terms of the quantities,
and sketch graphs showing
key features given a verbal
description of the relationship.
Key features include:
intercepts; intervals where
the function is increasing,
decreasing, positive, or
negative; relative maximums
and minimums; symmetries;
end behavior; and periodicity.*
2. We use the intersect feature of the calculator to
determine the coordinates of the vertices. Starting with
the upper left vertex and moving counterclockwise
around the region, they are:
The intersection of Y2 and Y3:
The intersection of Y1 and Y3:
The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
A(4.38, 45.62)
B(8.52, 23.48)
Ch 8–11
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
5V[LZ
Chapter 8:
Rational Functions
Interpret functions that
arise in applications in
terms of the context
(F-IF.5) Relate the domain
of a function to its graph
and, where applicable, to
the quantitative relationship
it describes. For example, if
the function h(n) gives the
number of person-hours
it takes to assemble n
engines in a factory, then the
positive integers would be an
appropriate domain for the
function.*
Analyze functions using
different representations
(F-IF.7) Graph functions
expressed symbolically and
show key features of the
graph, by hand in simple
cases and using technology
for more complicated cases.*
d. (+) Graph rational
functions, identifying zeros
and asymptotes when
suitable factorizations are
available, and showing end
behavior.
The Charles A. Dana Center
at The University of Texas at Austin
Ch 8–12
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Paintings on a Wall
Chapter 8:
Rational Functions
Teacher Notes
Build a function that models
a relationship between two
quantities
(F-BF.1) Write a function
that describes a relationship
between two quantities.*
b. Combine standard function
types using arithmetic
operations. For example, build
a function that models the
temperature of a cooling body
by adding a constant function
to a decaying exponential, and
relate these functions to the
model.
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The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Ch 8–13
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
The intersection of
The intersection of
The intersection of
The intersection of
Y1
Y2
Y2
Y2
and Y3:
and Y3:
and Y4:
and Y4:
Chapter 8:
Rational Functions
C(23.48, 8.52)
D(45.62, 4.38)
E(36.18, 13.82)
F(13.82, 36.18)
To ¿QG the vertices algebraically, we solve the IROORZLQJ systems by VXEVWLWXWLRQ
This gives the x-coordinate for vertices B and C.
Substitute for x in Y1 to get the y-coordinates.
This gives the x-coordinate for vertices D and A.
Substitute for x in Y2 to get the y-coordinates.
The Charles A. Dana Center
at The University of Texas at Austin
Ch 8–14
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
This gives the x-coordinate for vertices E and F.
Substitute for x in Y2 to get the y-coordinates.
3. Remember that the coordinates (x, y) of the points in the plane represent (width, length)
of the painting. Therefore, we can consider points only in the ¿UVW quadrant.
a. The region where the perimeter and area are acceptable is the closed region
between the lines and the curves.
b. For the perimeter to be too short and the area acceptable, we need ¿UVW quadrant
points below the line y = 32 – x but between or on the curves.
The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Ch 8–15
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
For the the perimeter to be too long and the area acceptable, we need ¿UVW quadrant
points above the line y = 50 – x but between or on the curves.
c. For the perimeter to be acceptable and the area too VPDOO we need ¿UVW quadrant SRLQWV
between or on the lines but below the curve y The Charles A. Dana Center
at The University of Texas at Austin
Ch 8–16
200
.
x
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Paintings on a Wall
Chapter 8:
Rational Functions
Teacher Notes
For the perimeter to be acceptable and the area too large, we need ¿UVW quadrant SRLQWV
between or on the lines but above the curve y 500 .
x
d. If the perimeter and area are both unacceptable, we need ¿UVW quadrant SRLQWV both
outside the lines and the curves.
The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Ch 8–17
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
Extension Questions:
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To minimize both perimeter and area you must be at the intersection of
Y1 32 x and Y3 200
x
Y2 50 x and Y4 500
x EHFDXVHWKHVHJLYHWKHXSSHUOLPLWVRQSHULPHWHUDQGDUHD
because these give the lower limits on perimeter and
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UHVSRQVHVWRtheSUHYLRXVWZRTXHVWLRQV"
The restrictions on the perimeter stay the same, but we must replace area restrictions
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The Charles A. Dana Center
at The University of Texas at Austin
Ch 8–18
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
/HWG WKHOHQJWKRIWKHSDLQWLQJ¶VGLDJRQDOLQLQFKHV7KHQVLQFHd 2 l 2 w 2 ,
25 b d b 40  252 b d 2 b 402
252 b l 2 w 2 b 402
252 w 2 b l 2 b 402 w 2
252 w 2 b l b 402 w 2
The area boundary equations are replaced with
l 252 w 2 and l 402 w 2
7KHJUDSKRIWKHUHJLRQUHSUHVHQWLQJDFFHSWDEOHGLPHQVLRQVLVWKH¿UVWTXDGUDQWUHJLRQ
EHWZHHQWKHOLQHVDQGEHWZHHQWKHVHPLFLUFOHV,WGRHVQRWLQFOXGHSRLQWVRQWKHD[HV
VLQFHWKDWZRXOGJLYH\RX]HURZLGWKRU]HUROHQJWK
7R¿QGZKHUHWKHSHULPHWHUOLQHVDQGGLDJRQDOFLUFOHVLQWHUVHFWZHVROYHWKHV\VWHPV
consisting of Y1 and Y3 and of Y2 and Y4)RUH[DPSOH
Y1 32 x
Y3 252 x 2
Let Y1 Y3
32 x 252 x 2
x 2 64x 322 252 x 2
2x 2 64x 322 252 0
$SSO\WKHTXDGUDWLFIRUPXODWRJHW[ LQFKHVRU[ LQFKHV
The Charles A. Dana Center
at The University of Texas at Austin
*OHW[LY!9H[PVUHS-\UJ[PVUZ
Ch 8–19
Algebra Assessments Through
the Common Core (grades 6-12), 2011
Paintings on a Wall
Teacher Notes
Chapter 8:
Rational Functions
Similarly, we solve for the intersection of Y2 and Y4 to get x = 11.77 inches or x = 38.23
inches.
The vertex points are (8.48 , 23.52), (23.52, 8.48), (11.77, 38.23) and (38.23, 11.77).
The Charles A. Dana Center
at The University of Texas at Austin
Ch 8–20
Algebra Assessments Through
the Common Core (grades 6-12), 2011
*OHW[LY!9H[PVUHS-\UJ[PVUZ