Balanced Assessment Test –Algebra 2008
Core Idea
Task
Score
Expressions
Representations
This task asks students find algebraic expressions for area and perimeter of
parallelograms and trapezoids. Successful students could show how the formula for
area of a trapezoid is derived from the area of the two triangles made by decomposing
the shape.
Buying Chips and Candy
Algebra
This task asks students to form and solve a pair of linear equations in a practical
situation. Successful students could use substitution or systems of equations to find
their solutions.
Sorting Functions
Representations
This task asks students to find relationships between graphs, equations, tables and
rules. Successful students could describe how to look at an equation and predict the
shape of the graph.
Sidewalk Patterns
Functions
This task asks students to work with patterns and find the nth term of a sequence.
Successful students could write an equation to finding the nth term.
Functions
Functions and
Representations
This task asks students to work with graphs and equations of linear and non-linear
functions. Students need to identify points on a graph, write a linear equation.
Successful students knew the difference between quadratic and exponential equations
and could give the equation of a parabola.
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Expressions
This problem gives you the chance to:
• work with algebraic expressions for areas and perimeters of parallelograms and
trapezoids
a
1. Here is a parallelogram.
h
b
The area of a parallelogram is
the product of its base times the
perpendicular height.
a. Which of these are correct expressions for the area of this parallelogram?
Draw a circle around any that are correct.
ab
!
!
1
ab
2
1
ah
2
ah
2a + 2b
2(a + b)
abh
b. Which of these are correct expressions for the perimeter of the parallelogram?
!
!
!
!
Draw!a circle around any!that are correct.
1
1
ab
ah
2a + 2b
abh
ab
ah
2(a + b)
2
2
2. Here is a trapezoid. It is made up of two triangles, each with height h.
a
!
!
!
!
!
!
h
b
Find the area of each of the two triangles and use your results to show that the area of the
1
trapezoid is ( a + b) h .
2
!
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4
Expressions
Rubric
The core elements of performance required by this task are:
• work with algebraic expressions for areas and perimeters of parallelograms and
trapezoids
Based on these, credit for specific aspects of performance should be assigned as follows
1.a Gives correct answer: ah circled and no others circled
b.
section
points points
1
Gives correct answers: 2a + 2b and 2(a + b)
2x1
Deduct 1 point for 1 extra and 2 points for more than 1 extra.
2.
3
Provides a convincing development of the required expression such as:
Shows the areas of the two triangles are
Adds these two expressions to get
!
1
1
ah and bh
2
2
1
(a + b)h
2
!
2x1
1
Total Points
3
6
!
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Expressions
Work the task and look at the rubric. What are the algebra tools a student needs to know
to do this task? What would you like to see if for complete explanation or proof in part 3?
_______________________________________________________________________
Look at student work for part 1a, finding the area of a parallelogram. Remember the
formula is given in words. Now look at student work. How many of your students put:
ah
Omitted
ab
abh
1/2ab
1/2ah
2a +2b
2(a+b)
ah
What surprises you about these results? How often do students get to work with algebra
in the context of geometry? How might this context help students to see algebra as a
sense-making tool?
Now look at student work for part 1b, finding the perimeter of a parallelogram. How
many of your students put:
Both
Omit
Omit
ab
1/2ab
ah
1/2ah
abh
formulas
2a+2b
2(a+b)
Aside from the geometry, how many students didn’t see the equivalency between 2a+2b
and 2(a+b)? Why do you think this was difficult for students? Why do you think students
had difficulty expressing perimeter in a geometric setting?
Now look at work for part 2. How many students could complete the entire argument? In
their work, how might you encourage them to improve their answers, make them clearer
and highlight the mathematics and logic of each step?
Now look at types of errors. How many students:
• Made no attempt on this part of the task____________________
• Put either 1/2 a or 1/2 b as the area of a small triangle__________
• Tried to use numbers instead of variables to solve the problem__________
• Found the area of the small triangles but didn’t combine them to try and complete
the argument or proof________________
• Other errors ___________________
How often do students in your class get the opportunity to make and test conjectures
about geometric or other contexts using algebra? What are some of your favorite
problems?
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Looking at Student Work on Expressions
Student A is able to identify the expressions for finding area and perimeter of a
parallelogram. The student is able to see that 2a+2b is equivalent to 2(a+b). In part 2,
Student A labels the two triangles in order to define which areas are being found by each
area expression. Then the student uses words and symbols to discuss combining the two
separate areas into a single expression.
Student A
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Student B uses diagrams to think about the expressions for area of each triangle. Then the
student talks about combining the two expressions. Student B factors out first the height
and then the 1/2 to make the combined expression equivalent to the original formula.
Understanding how to close an argument and show the steps back to the original
statement is an important piece of logical reasoning.
Student B
Student C has all the information needed to make the conclusion, but either doesn’t
understand how to factor out the expression or recognize a need to make the final
statement the same as the original.
Student C
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Student D finds the area of each separate triangle, but does not combine the terms or
make any attempt to show how that relates to the original formula.
Student D
Student E finds the area of the separate triangles and then seems to try to work backward
from the original formula to get the two expressions. The student appears to be
attempting distributive property on the right, but does not carry it through correctly.
Student E
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Student F might be debating between two different area formulas, using one strategy for
triangle a and a different strategy for triangle b. The students seems to settle on ah + bh.
The student then factors this expression but does not know how to get the 1/2 into the
problem. If you could interview this student, what question might you want to ask? If you
could pose some other problems, what might help you understand where the thinking or
skills break down? Notice that student F puts multiple choices for area in part 1a and
doesn’t recognize equivalent expressions for perimeter in part 1b.
Student F
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Many students struggled with using the distributive property on the original formula.
Student G distributes the 1/2 on variables inside and outside the parentheses. Student H
distributes the 1/2 over both variables and then separately distributes the h.
Student G
Student H
Student I struggles with the concept of like and unlike terms. The student also tries to
replace the variables with numbers to check the solution.
Student I
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Student J tries to use distributive property inappropriately and also teases apart the
expression in an attempt to set up an equation. On the right, the student also seems to use
parentheses more in a linguistic sense to separate things rather than as a mathematical
notation.
Student J
Student K can’t think with variables. To identify the expressions in 1a and b, the student
needs to put in numbers to think through the process and then substitute back in the
variables. In part 2 the student checks out one case of using numbers to check that the
formula is true, but doesn’t have the sense of the importance of variables to build a
generalizable proof for all cases. How do we help students learn to think with variables?
How do we help develop in students the idea of algebra as a tool for generalization
instead of set of manipulations?
Student K
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Algebra
Task 1
Expressions
Student Task
Work with algebraic expressions for areas and perimeters of
parallelograms and trapezoids.
Core Idea 3
Represent and analyze mathematical situations and structures
Algebraic
using algebraic symbols.
Properties and
• Use symbolic algebra to represent and explain mathematical
Representations
relationships.
Mathematics of this task:
• Using variables to find area and perimeter of a parallelogram
• Recognizing equivalent expressions by factoring or using distributive property
• Using algebra to make and prove a generalization
• Explaining the steps of factoring or using distributive property to make two
expressions equivalent
Based on teacher observations, this is what algebra students know and are able to do:
• Students were able to recognize expressions for finding perimeter of a
parallelogram
• Many students could recognize equivalent expressions for perimeter
Areas of difficulty for algebra students:
• Finding the area of a parallelogram/ translating from words to variables (the
formula for area was given in a verbal form)
• Thinking with variables instead of numbers
• Decomposing the trapezoid into two triangles
• Using factoring and/or distributive.0 property to make equivalent expressions
• Using algebra to make a generalization
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The maximum score available for this task is 6 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Many students, 80%, could find one expression for the perimeter of a parallelogram.
More than half the students could either find one expression for area and perimeter of a
parallelogram or find two expressions for a parallelogram. Some students, about 31%,
could find two expressions for perimeter of a parallelogram, and find the area of the two
small triangles. 13% of the students could meet all the demands of the task including
finding an expression for the area of a parallelogram and using algebra to show how to
combine and factor the expressions for the area of two triangles into the formula for the
area of a trapezoid.
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Expressions
Points
Understandings
Misunderstandings
82% of the students with this Students chose too many expressions for
0
score attempted the task.
2
3
4
5
6
perimeter and didn’t see equivalent expressions.
26% of the students omitted 2a+2b for
perimeter. 31% omitted 2(a+b) for perimeter.
About 5% of the students chose each of the
following options:ah, 1/ah, ahb, ab, and 1/2 ab.
Students with this score
Students had difficulty identifying the formula
could either find both
for area, even though the verbal rule was given.
expressions for perimeter for While many students picked ah, they often
a parallelogram or find one
picked other choices as well. 35% of the
expression for area and one
students did not pick ah as one of the choices.
expression for perimeter.
Almost 14% of the students picked ab for area,
confusing side length with height. 14% picked
1/2ah and 12% picked 1/2ab for the area
formula, confusing area of a triangle with area
for a parallelogram. About 5% picked each of
the perimeter formulas for area.
Students could identify
Students had difficulty trying to make an
expressions for area and
algebraic generalization about area of a
perimeter.
trapezoid. Some students did not decompose the
shape into two triangles. 10% of the students
substituted numbers for variables. 8% left the
height out of the formula for the small triangles
(1/2 a or 1/2 b). 10% found the area of the small
triangles, but did no further work to show how
to combine the areas and create an equivalent
expression to the given formula.
Students with this score
Students applied algebra inappropriately:
could find the expressions
combining unlike terms, not being able to factor
for perimeter and the area of expressions, using distributive property
the small triangles in part 2
incorrectly.
or find one perimeter
expression and explain all of
part 2.
Students couldn’t or didn’t combine the areas of
the small triangles to make a complete argument
in part 2.
Students could use a
geometric context to write
expressions with variables
for area and perimeter of a
parallelogram and a
trapezoid.
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Implications for Instruction
Students at this grade level have been working with area and perimeter of parallelograms
since 4th grade and area of triangles since 5th grade. At this grade level students need to be
able to move from the specific solution to generalizing about why the formula works and
where it comes from.
Students in algebra have learned a variety of tools, such as using a variable to stand for a
side of any length, combining like terms, or factoring polynomials. The purpose of these
procedures or tools is to be able to prove numerical relationships or why something
works or discover under what circumstances it won’t work.
Students need to be exposed to a variety of contexts for applying their skills to make and
test conjectures, to prove mathematical relationships. Procedures that can’t be transferred
to new situations are not useful and will probably have a short retention rate.
Some of the most interesting data from algebra shows that students on cumulative tests
do best on the most recently taught topic, rather than building from foundational
knowledge at the beginning of a course. Students need more opportunities to connect
their thinking to its use in context.
Ideas for Action Research – Building Logical Arguments
Students, even at very young ages, are capable of learning the reasoning chains to make
logical proofs. They just need to be pushed with good questioning strategies. In the book,
Thinking Mathematically, Integrating Arithmetic and Algebra in Elementary School, the
authors explore classroom experiences with children in 2nd, 3rd, and 4th grades using
variables and learning the logic of proof. In the chapter on justification and proof,
second-grader Susie is able to use number properties to justify why –5 – (-5) + 5= 5. She
is first able to write that a + b – b = a. She justifies it by saying that any number minus
itself equals 0, so b-b = 0. Then any number + 0 equals itself. Finally she is able to say
that if both of these statements are true the original will also be true. “It is productive to
ask children whether their conjectures are always true and how they know that they are
always true for all numbers…..We consistently have been surprised at what children are
capable of when given the opportunity.” This book comes with a video of students
making and testing conjectures. Some of the videos might be useful in the classroom to
give students models of how to use and discuss mathematics.
Fostering Algebraic Thinking offers some grade level appropriate activities to help
students build their capacity for generalization and making proofs. Consider the problem
of finding combinations of consecutive numbers to make the different answers of 1 to 35.
After students have explored this problem they might be asked to describe patterns that
they found with the consecutive numbers. Students might provide a range of solutions
from all the numbers made of 3 consecutive numbers can be divided by three to a number
N is a consecutive sum of m numbers if m divides evenly into N and m is an odd number.
This resource offers students many intriguing problems to work on their abilities to make
logical arguments and develop proofs using variables. It also offers suggestions on types
of questions that help students build habits of mind that lead them to make better
justifications.
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A related MARS task is the Sum of Two Squares, Course 2- 2005, available on the
Noyce website. It starts with the premise from Lewis Carroll, that 2(x2 + y2) is always the
sum of two squares where x and y are a pair of non-zero integers. Students are given
opportunities to investigate the conjecture with numbers, then describe the relationship in
words and finally challenged to prove why this is always true using algebra. The
discussion on this task would be a good place for practicing the types of questioning
strategies suggested in the two references above.
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Buying Chips and Candy
This problem gives you the chance to:
• form and solve a pair of linear equations in a practical situation
Ralph and Jody go to the shop to buy potato chips and candy bars.
Ralph buys 3 bags of potato chips and 4 candy bars. He spends $3.75.
Jody buys 4 bags of potato chips and 2 candy bars. She spends $3.00.
Later Clancy joins Ralph and Jody and asks to buy one bag of potato chips and one candy
bar from them. They need to work out how much he should pay.
Ralph writes
3p + 4b = 375
1. If p stands for the cost, in cents, of a bag of potato chips and b stands for the cost, in
cents, of a candy bar, what does the 375 in Ralph’s equation mean?
2. Write a similar equation, using p and b, for the items Jody bought.
__________________________________
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3. Use the two equations to figure out the price of a bag of potato chips and the price of a
candy bar.
Potato chips________________
Candy bar _______________
Show your work.
4. Clancy has just $1. Does he have enough money to buy a bag of potato chips and a
candy bar?
____________
Explain your answer by showing your calculation.
7
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Buying Chips and Candy
Rubric
The core elements of performance required by this task are:
• form and solve a pair of linear equations in a practical situation
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives a correct explanation such as:
It stands for the 375 cents that Ralph spent. (Must have correct units)
points
section
points
1
1
2.
Writes a correct equation such as: 4p + 2b = 300
2
Partial credit
For an almost correct equation. ( Left hand side of equation must be correct)
(1)
2
3.
Gives correct answers: 45 cents or $0.45
and
60 cents or $0.60
1 ft
Shows correct work such as:
8p + 4b = 6
subtract 3p + 4b = 375
5p
= 225
P = 45
4 x 45 + 2b = 300
2b =120
b = 60
2 ft
Partial credit
For some correct work.
4.
(1 ft)
Gives a correct answer: no
and
Shows a correct calculation such as:
0.60 + 0.45 = 1.05
1 ft
1
Total Points
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7
20
Buying Chips and Candy
Work the task and look at the rubric. What are the big mathematical ideas in this task?
What does a student need to understand about algebra to be successful on this task?
In part 1, the prompt is hoping to get students to explain that the decimal number has
been multiplied by 100 or changed to a whole, converting it from dollars to cents.
However students had difficulty with the prompt and gave an explanation about the
meaning of the money in the context of the problem.
In part 3, students are asked to use the 2 equations to find the cost of chips and candy.
Looking at student work there was not much commonality in the incorrect answers, so
instead look at the strategies students used. How many of your students used:
Strategy
Led to correct
Led to incorrect
solution
solution
Guess and Check
Substitution
Multiplication/Addition Method
Add both equations without
multiplication (7p+6b=675)
Put equations but showed no work
Made false equivalencies
No work
Added in extra equations
How did this help you think about what students understand about algebra?
• What are some of the things you need to think about to be successful at
substitution or to make substitution more convenient?
• What do you have to understand to use the multiplication/addition method?
• Why do you think so many students received incorrect solutions using guess and
check? What ideas did they not understand?
• What algebraic concepts need to be understood to apply guess and check
successfully?
Part 4 required students to use their solutions in #3 to see if they added to more than a
dollar. How many of your students:
• Did not attempt this part of the task?
• Gave an answer that wasn’t supported with a quantitative reason? (They didn’t
total the 2 numbers.)
• Made arithmetic errors?
• Explained something else?
What do you students understand about justifying an answer? What types of activities
help them to internalize what is needed for a good response?
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Looking at Student Work on Buying Chips and Candy
Student A is able to identify the change in units in part 1 and give the correct equation for
part 2. The student uses the multiplication/addition strategy for solving 2 equations with
2 unknowns. Notice that the student uses substitution into both equations to check the
solutions. In part 4 Student A uses definition of givens and mathematical notation to
support the conclusion that there isn’t enough money for the chips.
Student A
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Student B solves the 4p +2b = 300 for b, then uses substitution to solve for p in the
second equation.
Student B
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In general students who attempted substitution were less likely to get a correct solution to
part 3, solving 2 equations for 2 unknowns. Student C seems to first try adding the two
equations together, which doesn’t help solve the problem. Then at the top right the
student makes a first attempt at using substitution, but chooses the equation and variable
that comes out to a fractional quantity. Then the student chooses the equation to solve for
b, which would have allowed her to use substitution. At this point the student seems to
switch to the multiplication/addition strategy and solve the problem successfully. This is
a nice glimpse into the thought process of someone just starting to learn new material and
the problem-solving necessary to see which algebraic tool to use. Success is possible
because of the habit of mind of perseverance and belief in one’s own ability to figure
things out. How do we foster this diligence in other students?
Student C
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Student D is an example of a successful student using substitution with a fractional
quantity. While this strategy is completed successfully, a good problem solving strategy is
to think about which equation is easiest to solve for. Which variable might yield whole
number solutions? How do we help students develop these types of internal questions to
guide them when confronted with problem situations?
Student D
Student E attempts to use substitution to solve for 2 equations with 2 unknowns.
However, there is no evidence of using that information to complete the process. The
values for potato chips and candy bars could just as easily have come from guess and
check. There is major information missing from the solution.
Student E
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Student F also attempts to use substitution, picking the more complicated equation to
solve for first. The student tries to avoid fractions and in the process loses the exact
solution due to rounding issues. How might you pose this as a question to the class to let
them see and understand the potential errors caused by using decimal values? What do
you want them to understand about this solution set?
Student F
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Student G adds the two equations together in their original format. This doesn’t
eliminate either variable and so the student struggles with how to solve the resulting
equation. The student then seems to revert to the more comfortable process of guess and
check to get the solution set.
Student G
Student H also adds the equations and then uses an incorrect procedure to arrive at a
solution. What are some of the basic principles of solving equations that this student is
missing? What habits of mind might have helped the student see her error in thinking?
Student H
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Student I is struggling with many of the basic ideas of algebra. In part one the student
misunderstands the given equation, thinking it is finding number of items purchased
rather than cost. The student doesn’t seem to use the given information to write an
equation, but makes up his own. In part 3 the student tries to move items across the equal
sign to make the equation easier to solve. The student misses the negative sign and
makes other errors in trying to use equality or balance principles. The student then uses
guess and check to find a solution to the equation. However there is no attempt to relate it
to the other information given in the problem. The student doesn’t understand that the
equation is a function with multiple solutions and that more information is needed to
narrow it down to one unique solution.
Student I
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While Student J is able to explain that the 375 represents the total, the work in the
equation shows that the student has noticed the change from dollars to cents. In fact the
student also uses incorrect monetary notation in the solution to part 3. The student
attempts to solve the equation by manipulating the format of the equation, moving the 2b
to the right and then dividing both sides by 4. So the student has memorized an algorithm
for solving equations. However the student doesn’t know how to divide the –2b by 4 or
the –4p by 2. The student just ignores the complicating extra variable in a need to reach a
solution. What are some of the fundamental principles of variables and equations that
this student is missing? How do we make these big algebraic concepts more explicit and
visible in the classroom?
Student J
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Student K doesn’t understand equality and its relationship to formulating equations. This
student may still be thinking about the equal sign as “the answer follows”. The student
sets up 4 things as equal to each other, including 300 = 375. What other errors or
misunderstandings do you see in the solution process?
Student K
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Algebra
Student Task
Core Idea 3
Algebraic
Properties and
Representations
Task 2
Buying Chips and Candy
Form and solve a pair of linear equations in a practical situation.
Represent and analyze mathematical situations and structures
using algebraic symbols.
• Use symbolic algebra to represent and explain mathematical
relationships.
• Judge the meaning, utility, and reasonableness of results of
symbolic manipulation.
Mathematics of the task:
• Understanding equalities and maintaining equalities to set up and solve equations
• Defining variables
• Combining like terms, not combining unlike terms
• Understanding monetary units
• Understanding that two linear equations with two unknowns has a unique
solution, but that a linear function with two variables has an infinite number of
solutions
• Strategies or procedures for solving sets of equations: substitution,
multiplication/addition
Based on teacher observations, this is what algebra students know and are able to do:
• Write an equation from context and understand what the variables represent
• Use guess and check successfully as a strategy to find a solution for two equations
with two unknowns.
• Use substitution or multiplication/ addition to solve for the unknowns
Areas of difficulty for algebra students:
• Quantifying answers to justify a conjecture
• Using distributive property correctly in solving a problem
• Identifying which equation and which unknown would be easiest to use when
applying the substitution method
• Using partially remembered strategies, but could not carry them through the entire
solution process or unsuccessfully combined strategies
• Checking work with both equations to see if the solution is true for both
• Understanding that functions have multiple solutions
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The maximum score available for this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 86%, could determine how to write an equation in part 2. Many students,
68%, could write an equation and explain that the units had changed from dollars to cents
or determine whether there was enough money to buy the chips and candy. A little less
than half the students could write the equation, determine the units, and write a
justification about having or not having enough money to purchase the snacks. 14% of
the students could use algebra to solve two equations with two unknowns, using
substitution or multiplication/addition. 8% of the students scored no points on this task.
80% of the students with this score attempted the task.
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Buying Chips and Candy
Points
0
Understandings
80% of the student with this
score attempted the task.
2
Students could write an
equation from a verbal
description.
3
Students could write an
equation and either explain the
units as cents or write a
convincing statement about
having enough money.
Students could explain the
units in the equations, write an
equation, and determine if
there was sufficient money to
buy the snacks.
4
5
7
Students could solve all of the
task, but used guess and check
for the solution strategy in part
3.
Students could solve two
equations with two unknowns,
using algebra. Students could
talk about the context of the
problem in terms of the
variables and numbers in the
equation. Students could make
a justification by quantifying
supporting information.
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Misunderstandings
Students had difficulty writing the
equation. About 4% did not attempt the
equation. 4% put a decimal point in the
equation.
Students had difficulty explaining the units
in part 1. 20% explained the meaning of the
number as total cost. Another 10%
explained it as how much he pays.
Almost 20% of the students did not attempt
part 4 of the task, even though it just
involved arithmetic. 11% gave correct
answers, but didn’t quantify their answers.
Students did not know how to solve
equations for two unknowns. 17%
attempted substitution. 15% attempted
multiplication/addition. 7% just added
together the two equations, which did not
eliminate one of the variables. 6% wrote
the two equations but showed no work.
15% of the students did not answer part 3.
14% had answers with 75 cents as one of
the two costs. 3% had incorrect monetary
notation. Many students used distributive
property incorrectly.
23% of all students attempted guess and
check. Many were not successful, because
they didn’t check their answers against all
the constraints.
31
Implications for Instruction
Students at this grade level need to learn mathematical ideas in context. They need to see
a purpose or application for the various procedures that they are being shown. Within
every lesson or set of lessons, students need to move from the concrete, to pictorial, to the
abstract. It does students no good to be able to solve two equations with two unknowns,
if they never think to use that procedure when confronted with a problem situation.
Students need to be able to think through the design of the procedure, why does it work.
Many students attempted to use a procedure, but could not remember the correct
sequence of steps or forgot steps. In struggling with the justification of the procedure or
algorithm, students make sense of the operations and have resources for thinking through
what needs to be done when they forget.
Students need to also develop habits of mind for checking solutions and sense-making
around their answers. For example, many students found solutions that would work
correctly in one of the equations, but not in both of the equations. Students need
opportunities to discuss solution strategies. Which equation or variable would it be easier
to solve for if using a substitution strategy? Why? Which equation or variable would it
be easier to solve for if using a multiplication and subtraction of equations strategy?
Why?
Students need to move from just learning procedures to thinking about the meaning of
functions. Why does an equation with 2 unknowns have an infinite number of solutions?
What is meant by “variable”? They need to see and discuss in context how the variation
in one of the unknowns also causes variation in the other and why. Students also need to
see and discuss why having 2 equations with 2 unknowns produces only one unique
solution. Why did the second equation narrow the range of possible solutions? They
should be able to talk about this meaning within the context of a situation, within the
context of a graphical representation, as well as showing the numerical answer.
With the ease of calculation afforded by calculators, students need to work with a variety
of solutions not limited to just whole numbers. Some students show evidence of giving
up or changing strategies when problems don’t come out evenly. They have made an
incorrect generalization based on the normal problems they work with than answers are
always whole numbers..
Ideas for Action Research: Looking at Student Work to Plan
Remediation:
Often when planning remediation or helping students who are behind, teachers think
about the students who are almost there. What are the few steps they need to be
successful? But what is it that the students who are at the lowest end of the spectrum
need? How are their issues different?
Sit down with colleagues and examine the following pieces of student work. Consider the
following questions:
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1. What are the strengths, if any, that the student has? What are the concepts the
students understand about the situation? How might these strengths be used to
help build their understanding of the whole situation?
2. How did students use representations? Were the representations accurate? Why
or why not? What would have helped the student to improve their representation?
3. What misunderstandings does the student have? What skills is the student
missing? What does this suggest about a specific course of action to help this
student?
4. How are the needs of each of these students the same or different?
After your have carefully looked at each piece of student work, see if you can devise a
plan of experiences/ discussions/ tools that might help these students to make more sense
of these situations. While you don’t have these exact students in your class, each
member of the group will probably have students with similar misunderstandings.
Identify students who you think are low and plan different approaches for attacking the
problems outlined here. Have each person in the group try out a different course of action
and report back on the how the lesson or series of lessons effected the targeted students.
See if you can all use some similar starting problems and bring work of the students to
share. What types of activities or experiences made the most noticeable improvement in
student work?
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Andy
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Barbara
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Cal
Cal did not attempt part 4.
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Debbie
While Debbie did some work on page one of the task it was all up in the prompt area. In
part 3 Debbie again does some calculations, but gives no answer for part 3 or 4.
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Ed
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Farrah
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Sorting Functions
This problem gives you the chance to:
• Find relationships between graphs, equations, tables and rules
• Explain your reasons
On the next page are four graphs, four equations, four tables, and four rules.
Your task is to match each graph with an equation, a table and a rule.
1. Write your answers in the following table.
Graph
Equation
Table
Rule
A
B
C
D
2. Explain how you matched each of the four graphs to its equation.
Graph A ___________________________________________________________
___________________________________________________________________
___________________________________________________________________
Graph B ___________________________________________________________
___________________________________________________________________
_________________________________________________________
Graph C __________________________________________________________
___________________________________________________________________
___________________________________________________________________
Graph D ________________________________________________
___________________________________________________________________
____________________________________________________________________
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Graph A
Equation A
Table A
y is the
same as x
multiplied
by x
xy = 2
Graph B
Equation B
Table B
Equation C
Table C
Equation D
y=x-2
Rule C
y is 2 less
than x
y = x2
Graph D
Rule B
x
multiplied
by y is
equal to 2
y2 = x
Graph C
Rule A
Table D
Rule D
x is the
same as y
multiplied
by y
10
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Rubric
Sorting Functions
The core elements of performance required by this task are:
• find relationships between graphs, equations, tables and rules
• explain your reasons
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1 Gives correct answers:
.
3
(2)
(1)
3
(2)
(1)
Equations: 4 correct 3 points
points
3 or 2 correct 2 points
point
1 correct 1 point
Table: 4 correct 3 points
Rule: 4 correct 2
3 or 2 correct 2 points
3 or 2 correct 1
2
(1)
8
1 correct 1 point
2 Gives correct explanation such as:
. Graph A is a parabola/quadratic curve that passes through the origin
and is symmetrical about the y axis (every value of y matches two
values of x that are equal in size with opposite signs), so its equation
is y = x2.
Graph B is a straight line, so its equation is linear, y = x – 2.
Graph C is a parabola that is symmetrical about the x axis (every
value of x matches two values of y that are equal in size with
opposite signs), so its equation is x = y2.
Graph D: If we take any point on the graph and multiply its coordinates, say, (2, 1), we get 2. This is the equation xy = 2. Accept,
we have matched the other three graphs to equations.
Accept alternative correct explanations
Partial credit
2 or 3 correct explanations
Total Points
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(1)
2
10
42
Sorting Functions
Work the task and look at the rubric. What important algebraic ideas might students use to match a
graph with an equation? What connections do you hope students are making to relate this
information?_____________________________________________________
Look at student work on matching the representations. In general did students have more difficulty
with equations, tables or rules. Use this table to help you chart the information.
Graph
A
B
C
D
Equation
C
D
B
A
Table
B
A
C
D
Rule
A
C
D
B
What surprised you as you charted the information? What seems most difficult for students to
understand? What types of experiences or questions do students need to have to help them develop
these big ideas?
Now look at the student explanations for part 2. How many of your students:
• Could use correct algebraic ideas to think about the shapes of the graphs and the
corresponding types of equations_______________
• Only talked about matching graphs to tables ___________________
Make a list of some of your best explanations. How could you use these as models or to pose
questions for discussion to help other students develop the logic of justification?
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Looking at Student Work on Sorting Functions
Here are the results of students work on the table. Many students did not even understand the logic
of sorting and put A, B, C, and D for each choice. That is harder to capture in the data.
Graph
A
B
C
D
Equation
Table
Rule
Bold = correct response
Italic= error choice for each response
C
A
B
D
B
A
C
D
A
B
C
D
8% 15% 3%
6% 9% 10%
8% 1% 10%
D
A
B
C
A
B
C
D
C
A
B
D
7% 9% 3%
5% 5% 4%
3% 8% 6%
B
A
C
D
C
B
C
D
D
A
B
C
8% 20% 4%
8% 1% 9%
10% 10% 8%
A
B
C
D
D
A
B
C
B
A
C
D
9% 4% 11%
5% 4% 5%
6% 8% 12%
The second table just summarized the percent of students making errors for each part.
Graph
A
B
C
D
C
D
B
A
Equation
26%
19%
32%
24%
B
A
C
D
Table
25%
14%
18%
14%
A
C
D
B
Rule
19%
17%
28%
26%
Students had a very difficult time giving reasons for matching graphs to equations. Between 14 to
20% of the students gave no response to each part of question 2. About 34% of all students just gave
the vague explanation of matching graph to table and then find the equation. However some students
brought out some very interesting and useful algebraic concepts to think about how to match the
information. How do we help students make connections between algebraic concepts and move
beyond procedural knowledge? An important piece of algebraic thinking is to move from a specific
solution to making generalizations about types of solutions. What opportunities do we provide to
help students to think in a more global perspective? Here are a few examples of what algebra
students could do.
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Student A recognizes that equations with x2 will yield a parabola. The student uses several
properties of linear functions to explain graph B. Student makes connections between similarities
and differences in the graphs and equations of A and C. In part 4 the student explains why for this
equation there will be no y-intercept.
Student A
Student B makes a good case for why there is no y-intercept for Graph D.
Student B
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Student C uses some interesting language to describe the differences between Graph A and B,
giving more details about the parabolas. For graph D the student makes an argument about
symmetry. What further questions or investigations could you pose for students to help them learn
more about the parts of the equation that determine the symmetry or to explore how the symmetry of
this graph is different from the symmetry of the parabola?
Student C
Student D describes how to determine which parabola is equal to graph A by looking at intercepts.
The student also uses knowledge of intercepts to identify graph B.
Student D
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Student E uses information about the table and plotting graphs and goes into detail about they relate
to each other. Notice that the student solves for y for graph D to help make sense of the shape.
Student E
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Student F gives the minimum descriptions to get the points. What further questions or investigations
might you want to pose around the response to graph D?
Student F
Student G gives an implied elimination answer for graph D.
Student G
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Student H again uses the matching strategies but gives enough details to make it a valid explanation.
What experiences or questions might push this student’s thinking to the next level?
Student H
Student I is an example of a student whose responses are too vague for part C and D.
Student I
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Student J is able to think about parabolas and choose the correct representations, but struggles with
the language to explain or make generalizations about B and D.
Student J
For graph A, 87% of the students who got the explanation correct talked about parabolas. 8% talked
about quadratics.
For graph B, half the students who got the explanation correct talked about it being linear. 18%
talked about the equation being in the form of y=mx + b. 16% talked about the y-intercept = -2. 9%
talked about the slope.
For graph C students talked about a sideways or strange parabola.
For graph D, most students who got the explanation correct used an elimination argument. Some
students gave an explanation about why there was no y-intercept or the effects of multiplying by 0.
A few students used a symmetry argument, solving for y, or a hyperbola to make their point.
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Algebra
Task 3
Sorting Functions
Student Task Find relationships between graphs, equations, tables and rules.
Core Idea 1
Functions
and Relations
Core Idea 3
Algebraic
Properties and
Representations
Explain your reasons.
Understand patterns, relations, and functions.
• Understand relations and functions and select, convert flexibly
among, and use various representations for them.
Represent and analyze mathematical situations and structures
using algebraic symbols.
• Use symbolic algebra to represent and explain mathematical
relationships.
• Judge the meaning, utility, and reasonableness of results of
symbolic manipulation.
The mathematics of this task:
• Making connections between different algebraic representations: graphs, equations, verbal
rules, and tables
• Understanding how the equation determines the shape of the graph
• Developing a convincing argument using a variety of algebraic concepts
• Being able to move from specific solutions to thinking about generalizations
Based on teacher observations, this is what algebra students know and are able to do:
• Understand that squaring a variable yields a parabola and that the variable that is squared
effects the axis around which the parabola divided
• Use process of elimination as a strategy
• Match equations to tables and graphs
• Look for intercepts as a strategy
• Use vocabulary, such as: parabola, intercept, and linear
Areas of difficulty for algebra students:
• Knowing the difference between linear and non-linear equations
• Not knowing how to explain how they matched the graph and the equation
• Connecting the constant to the slope, e.g. just because it’s – 2 doesn’t meant it’s negative
slope
• Quantifying: even though they could describe the process, but didn’t quantify
• Not knowing how or when to use the term “curve” or parabola
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The maximum score available for this task is 10 points.
The minimum score for a level 3 response, meeting standards, is 6 points.
Most students, 83%, could match two or three correct graphs to the table. Many students 76% could
also match at least 1 graph to an equation. More than half the students, 56%, could match two or
three graphs to equations, tables, and rules. Almost half the students, 46%, could match correctly all
the representations. 14% could meet all the demands of the task including explaining in detail how
to match a graph to its equation using algebraic properties about graphs and equations. 13% of the
students scored no points on this task. 94% of the students with this score did not attempt the task.
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Sorting Equations
Points
0
2
3
6
8
10
Understandings
Misunderstandings
94% of the students with this score did not
attempt the task.
Students could match 2 or 3
See table in Looking at Student Work for
graphs with tables.
specific errors.
Students could match 1 or 2
See table in Looking at Student Work for
graphs with equations and tables. specific errors.
Students could match some graphs See table in Looking at Student Work for
with equations, tables, and rules.
specific errors.
Students had could match all the
Students had difficulty giving a complete
graphs with their equivalent
explanation of how to match a graph with
representations in the form of
an equation. Students gave vague
equations, tables, and verbal rules. explanations, such as matching the graph
with a table. Students were not thinking
about the general shapes of the graphs and
the general equations that form those
shapes.
Students could match graphs to
equations, tables, and verbal rules
and think in general terms about
how equations determine the
shape of graphs.
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Implications for Instruction
Students should be able to understand the relationship between equations, graphs, rules, and tables.
Students should know a variety of ways to check these relationships. Lessons should regularly focus
on relating multiple representations of the same idea. It is important that algebraic ideas not be
taught in isolated skill sets. Consider this quote from Fostering Algebraic Thinking by Mark
Driscoll, “ One defining feature of algebra is that it “introduces one to a set of tools – tables,
graphs, formulas, equations, arrays, identities, functional relations, and so on – that constitute a
substantial technology that can be used to discover and invent things. To master the use of these
tools, learners must first understand the associated representations and how to line them together.
A fluency in linking and translating among multiple represent seems to be critical in the
development of algebraic thinking. The learner who can, for a particular mathematical problem,
move fluidly among different mathematical representations has access to a perspective on the
mathematics in the problem that is greater than the perspective any one representation can
provide.”
Ideas for Action Research – Review of the Literature – Linking Multiple
Representations
Sometimes in the pressure to move through the curriculum, we as teachers rely too heavily on the
sequence provided by our textbooks. It is important to occasionally step back and think about the
subject as a whole and what are the important concepts we want students to develop. Consider
taking time to read and to discuss some professional literature with colleagues.
• What are the important ideas being presented?
• What are the implications for the classroom?
• How can we design some specific activities or lessons to fit into our program that will help
develop some of the ideas we have just read?
• Why is this important for students?
One interesting resource related to this task would be Chapter 7 – Linking Multiple Representations
from the book, Fostering Algebraic Thinking. Here are some key excerpts for consideration.
“Issues Regarding Understanding –
There are challenges in thinking algebraically that go beyond learning discrete pieces of
information. Often, difficulties can arise when it is assumed that students are attaching the same
meanings or making the same connections that are intended by the teacher.
1. Students may not see the links between different representations of a functional relation – for
example, the mutual dependence between a function’s graph and equation, or between its
table and equation.
2. Students may interpret graphs only point wise, not globally.
3. In the course of working on a problem, students may neglect to connect the representation
back to the original problem context.”
The chapter then goes on to give examples of classroom lessons that help develop this relational
thinking and interesting problems that can be used in the classroom.
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