Algebra
Core Idea
Mars 2006
Overview of Exam
Task
Task Descriptions
Score
Swimming Pool
Geometry and
Measurement
This task asks students to work with trapezoids, volume, rates and time graphs in the
context of a swimming pool. Successful students reason about rates per second to find
the total time to fill a swimming pool and choose a time/depth graph to match the
geometric situation of filling the pool. Students working at a high level could develop
a formula and calculate the volume of water in a swimming pool with two trapezoidal
sides.
Mathematical Reasoning Odd Sums
This task asks students to work with odd, even and consecutive numbers. Make and
justify conjectures about consecutive numbers. Successful students could give
examples of two consecutive numbers to make a given odd number or 3 consecutive
numbers to make an even number. Students were able to give a rule to determine if
an even number could be written as the sum of 3 consecutive numbers. Students
working at a high level could write a justification for why any odd number can be
written as the sum of two consecutive numbers.
Functions and Relations Patchwork Quilt
This task asks students to recognize and extend a number pattern for a geometric
pattern. Students expressed the rule using algebra and used inverse operations to solve
a problem. Successful students could identify and extend a pattern and write an
equation to show the pattern. Students could use their equations to solve the pattern
extensions of either variable.
Algebraic Properties and Printing Tickets
Representations
This task asks students to compare price plans using graphs and formulae. Use
inequalities in a practical context of buying tickets. Successful students were able to
write an equation to find the cost of buying tickets with an initial set up cost and
graph that equation. Students could look at a graph and use inequalities to determine
when to use different printing companies. Students working at a high level could use
two equations to solve for the break-even cost, when both printers would charge the
same.
Algebraic Properties and Graphs
Representations
This task asks students to relate line graphs to their equations. Successful students
could match key parts of graphs with their equations and write an equation of a line
that would pass through a given point.
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Swimming Pool
Rubric
The core elements of performance required by this task are:
• work with trapezoids, rates and time graphs in a real context
Based on these, credit for specific aspects of performance should be assigned as follows
1.
points
Gives correct answer: 9,900 cubic feet
1
Shows correct calculation such as: 60 x (8 + 3) x 30
2
1
section
points
2
2.
3.
Gives correct answer: 20 hours, 37.5 minutes
1
Shows correct calculation: dividing 74,250 by 60 x 60
= 20.625 hours
1
1
(a) Gives correct answer: Graph B
1
(b) Gives correct explanation such as:
At first the depth increases quickly, but then more slowly as the water
moves up the slope. For the final 3 feet, the depth increases at a
constant rate.
Partial credit
A partially correct explanation.
3
2
(1)
Total Points
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6
Algebra – Task 1: Swimming Pool
Work the task and examine the rubric.
What strategies might students use to find the volume of the swimming pool? See if
you can find at least two.
How did you have to decompose the shape of the pool to use each strategy?
Now look at the work of you students for finding the volume of the pool. How many
of your students put:
9900 43200 5400 74250 9000 1800 19800 14400 990 98 Other
What does the student know about the structure of the shape and volume for each of
these wrong answers? What were students confused about?
How often do students in your class get the opportunity to break apart geometric
figures?
How often do students in you class get the opportunity to justify where parts of a
formula come from or to derive their own formula?
Do students get opportunities to solve problems that involve adding lines or pieces not
present in the original diagram? How might this have helped students?
Did any of your students think about using the trapezoid as the base of the solid?
What is the mathematics involved in converting from seconds to hours? What
confused students about changing the 20.625 hours into hours and minutes? Look at
student work on the conversion. Can you sort their errors into categories?
Now look at student work on the graphs. How many of your students put:
B
B
A
A
C
D
With good With inc. Steady rate Shape of
explanation explanation
pool
Did students make the connection that the deep end would fill quickly and then slow
as the water moved up the slope of the pool?
Did students get confused about graphs, thinking that the shape of the graph should be
related to the shape of the pool? Give some examples of this type of description.
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Do you think some students didn’t understand how the pool would fill? (Some
students talked about filling in the shallow end first, then spilling over to the deep end
or filling as if the top filled first then the bottom.)
Why do you think this task was so difficult for students?
What types of experiences do students need with spatial visualization and
composing/decomposing shapes?
What types of experiences do students need with understanding graphs?
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Looking at Student Work on Swimming Pool:
There are three very different pieces of thinking needed to solve this task. For the first part, students
need to combine or synthesize lots of learning about geometry, 3-dimensional shapes, and finding
volume to derive a formula. For the second part, students need to do a conversion from seconds to
hours and minutes. To complete this part of the task, students need to understand the relationship
between a decimal quantity and a numerical value when decimal is parts of 60 instead of parts of
100. Again, students are being asked to synthesize previous knowledge to make sense of a less
familiar application. The final part is reasoning about a time graph by looking at a geometric shape
and thinking about how a pool would fill. For the first part of the task, students used four solution
paths:
Solution Path 1: Some students understand that the volume of a prism is the area of the base times
the height. To use this knowledge, the student needs to think of the sides of the pool, the trapezoids,
as the base and the width of the pool (30) as the height. The area of the trapezoid would be (8+3)/2
times 60. So the total volume would be (8+3)/2 x 60 x 30. See the work of Student A.
Notice that Student A is able to convert the decimal into minutes by changing the decimal to a
fraction and then using a proportion.
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Solution Path 2: This strategy involves adding extra lines or parts to the diagram to make more
familiar shapes. First the student makes the shape into a rectangular prism with the dimensions 8ft.
x 30 ft. x 60 ft., and finds the volume for this prism= 14,400 cu. ft. Then the student finds the
volume of the triangular prism added to the original pool, (1/2 x 5 x 60)(30) = 4500 cu.ft. Finally
the student subtracts the volume of the triangular prism from the volume of the rectangular prism.
See the work of Student B.
Student B
Notice how Student B uses labels to keep track of what’s known.
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Solution Path 3: This involves decomposing the pool into a top part, a rectangular prism, and a
lower part, a triangular prism, and add the volumes of the two parts together. Notice how Student C
draws in the lines to show the two parts he is thinking about.
Student C
Solution Path 4: Another way to solve this problem is to duplicate the pool and fit the two pieces
together to make a large rectangular prism with dimensions 11 x 30 x 60. This shape is easy to
calculate the volume for because it fits the standard formula,
w x h x l. Then the original shape is just half as much. This method involves good spatial
visualization.
Student D
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Now, look at some student work below. Where does their thinking break down? What are they not
understanding about composing/decomposing shape? What are they not understanding about
volume?
Student 1
Student 2
Student 3
Student 4
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Below are some examples of common error patterns by students in attempting to find volume.
Student E has internalized the volume formula from l x w x h to multiply all the numbers in the
figure. Student F combines dimensions and then multiplies. Student G finds the surface area of the
pool, not the volume. Student H confuses the gallons with the volume in cubic feet.
Student E
Student F
Student G
Student H
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Converting from seconds to hours proved challenging for students. Student I did not understand
standard time notation and did two calculations; one for finding hours and one for finding minutes.
Student I
Student J doesn’t know what to do with the decimal part of the number and just moves it over to the
minutes’ side.
Student J
Student K might have divided the decimal portion of the answer by 60 to get an additional hour
using the remainder as the minutes. What does this student not understand about the meaning of
decimals?
Student K
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Ideas about filling the pool and the graph:
B – Since the bottom of the pool is sloped, the water will fill faster near the bottom, then
slow as the water nears the 3 foot wall
A- Because the slope of graph A is consistent showing a steady rate
D- Represents the filling because of the way the tank was made. There is less space to fill in
the bottom making it faster than filling the top
Other confusions about filling the pool and graph: What might each student be thinking about?
C- because the pool is a trapezoid, so the water wouldn’t be very deep when it started filling
up, but it would be deeper as it reached the top.
A- because its going straight up from 8 feet to 3 feet just like the swimming pool
A – because it is a straight flat vertical surface
C – because it shows the depth increasing little by little as the pool gets deeper and deeper
C- As time goes on, the steady flow starts to fill up the deep end, making the pool deeper
C- When the pool fills up, it will get more water where the hose is lying so one side will
have more water since it’s deeper
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Algebra
Course One/Algebra
Student Task
Core Idea 4
Geometry &
Measurement
Core Idea 3
Alg. Properties
&
Representations
Core Idea 1
Functions and
Relations
Task 1
Swimming Pool
Work with trapezoids, volume, rates and time graphs in the context of
a swimming pool.
Understand measurable attributes of objects; and understand the
units, systems, and process of measurement.
•
Approximate and interpret rates of change, from graphic and
numeric data.
•
Analyze functions of one variable by investigating local and
global behavior, including slopes as rates of change, intercepts
and zeros.
Based on teacher observation, this is what algebra students knew and were able to do:
Students were able to convert from seconds to hours, but were unsure what to do with
the decimal
Areas of difficulty for algebra students:
Finding volume of an unfamiliar shape
Composing/ decomposing a shape into familiar parts
Confusing a state rate of water flow with a steady rise in the depth of the pool
Confusing the shape of the pool with the shape of the graph
Not recognizing that after 5 feet the depth would increase at a steady rate
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The maximum score available on this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Less than half the students, 43.7%, knew to divide by 360, to change seconds to hours, and could do
the calculation accurately. Some students, about 29%, could convert this answer to standard
notation of hours and minutes by successfully changing the decimal from 0.625hrs. to 37.5 minutes.
Only 13 % could make the conversion and then either find the volume of the pool or pick the correct
graph with a partial explanation of why it was correct. Less than 1% of the students could meet all
the demands of this task, including finding the volume of a trapezoidal prism and explaining how a
graph of time and depth matches the situation of water filling the pool. 43% of the students scored
no points on this task. 90% of the students with a score of zero attempted the task.
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Swimming Pool
Points
Understandings
90% of the students with this
0
2
3
5
8
Misunderstandings
Students did not understand how to
score attempted the task.
convert seconds to hours. Some students
only divided by 60. 11% did not attempt
to do the conversion. Students had
difficulty with decimal placement.
Students knew the process for
Students did not know what to do with the
conversions from seconds to hours numbers after the decimal point or did not
(divide by 3600) and could
understand standard notation.
calculate that accurately.
Students could successfully do all Students had difficulty choosing the right
the steps of converting from
graph. 60% of all students choose graph
seconds to hours and minutes.
A. 18% picked graph C. Less than 5%
picked graph D. Most students thought
filling at a constant rate would be a
straight line. Other students confused the
shape of the pool for the shape of graph.
Students could convert from
10% of the students multiplied all 4
seconds to hours and minutes and measures together to find the volume
either find the volume of the pool (43,200). 6% treated the pool as a
or pick the graph with a partial
rectangular prism with dimensions 8 x 30
explanation of why it was correct. x 60. 6% had an answer of 5400. Other
popular answers were 7200, 9000, and
101.
Students could
compose/decompose a 3dimensional shape into familiar
parts or add lines to make a
familiar shape. This helped
students to find volume of a
trapezoidal prism. Students could
convert from seconds to hours and
minutes using standard notation.
Students could reason about water
filling a swimming pool and
choose an appropriate time and
depth graph, explaining how the
shape of the graph matched the
context.
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Implications for Instruction
Students need more experience with spatial visualization, including composing and decomposing
geometric 2- and 3-dimensional shapes. By 7th and 8th grade students should start to work with
taking slices of 3-dimensional shapes and being able to draw and measure those slices. Students
need to think about rotations and flips of 3-dimensional shapes.
An important idea for solving geometric problems is the idea that lines can be added to 2dimensional shapes or that extra shapes can be combined with 3-dimensional shapes. This ability to
add on helps the problem solver find and use knowledge about more familiar shapes.
As students move into algebra, they should be pushed to generalize about geometric formulas; for
example, moving from volume of a rectangular prism is “l x w x h” to thinking about volume as the
area of the base times the height. This generalization can then apply to a wide variety of shapes, like
cylinders and triangular prisms. A large piece of algebraic thinking is developing mathematical
justification in words, diagrams, and symbols. Students should be able to connect the various
representations. Students, at the algebra level, should also be encouraged to justify why formulas
work; how do the various parts of the formula relate to the geometric context. Students should be
able to make a strong case for why, when finding the area of a triangle, the length times width is
divided by 2, or why when finding the area of a trapezoid the two bases are divided in two.
As students move through an algebra course they should have frequent experiences graphing
functions of real-life contexts, such as time/distance graphs. Students need to see that graphs
represent something different from the shape of object. For example, the graph of the height of a car
on a ferris wheel over time is not a circle, but a peak shape, with an steady increase and decrease.
Students need to discuss common misconceptions like this in order to see why these ideas are
incorrect. Possible activities might include explaining the story of a graph, or given a story make a
graph without the scale. Good examples can be found in the Language of Functions published by
the Shell Centre.
Action Research - Developing Justification, Connecting Geometric and Symbolic
Representations.
Have students work the MAC 4th Grade -Task2, 2004: Piles of Oranges or 5th grade 2001 Soup Cans.
Can your students develop a rule for finding any number in the pattern? Try to get them to use a
diagram to explain why the formula works and where the numbers come from.
Have students work the MAC Course1- 2000: Trapezoidal Numbers. See if students can find a
pattern for finding the number of dots for any number in the pattern. See if they can use diagrams to
explain why the formula works and where the numbers are represented in the diagram.
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Exploring Graphs
Consider the following problem, the vending machine. A school cafeteria contains a vending
machine, which sells sodas. On a typical day:
The machine starts half full
No drinks are sold before 9 am or after 4 pm
Drinks are sold at a slow rate throughout the day, except during lunch between 12:00-12:30
and after school from 3:00-3:30 when demand is greater
The machine is filled up just before lunch. (It takes 10 minutes to fill.)
Sketch a graph to show how the number of drinks in the machine might vary from 8 am to 6 pm.
See how students do with interpreting the various conditions for the graph. What do they seem
confused about. How did you facilitate the discussion to push their thinking about graphs?
Now have them work the MAC8th grade – 2006: Going To Town. Are they able to make sense of
the graph? What areas gave them difficulties?
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Algebra – 2006
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Algebra – 2006
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Odd Sums
Rubric
The core elements of performance required by this task are:
• work with odd, even and consecutive numbers
• make and explain justifications
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answers: 7 + 8
1
4.
2
Gives a correct explanation, perhaps involving a description of how the
sums can be made (‘subtract 1 from the number, divided the result by 2,
and that gives the smaller of the two consecutive numbers in the sum’)
Algebraically
All odd numbers can be written in the form 2n + 1, where n is an integer.
And 2n + 1 = n + (n +1)
3.
section
points
1
49 + 50
2.
points
2
2
Gives three correct examples.
2
Partial credit
Gives two correct examples.
(1)
Gives a correct explanation such as:
(An even number that is) a multiple of 3 can be written as the sum of three
consecutive whole numbers.
or
A multiple of 6 can be written as the sum of three consecutive whole
numbers.
2
2
or
2
Total Points
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8
23
Algebra – Task 2: Odd Sums
Work the task and examine the rubric.
Can you use algebra to show why every odd number can be written as the sum of
two consecutive whole numbers?
Can you use pictures, diagrams or justifications about number structure to show
why every odd number can be written as the sum of two consecutive whole
numbers?
Now look at the work of your students for part two. How many of your students:
Tried to
Attempted to Talked about Just stated
Gave a
use algebra
odds as
the rule “an
procedure for justify by
giving more
having an
even + an
finding the
examples
extra bit
odd = odd,
pair of
with no
numbers
completion
of the logic
What opportunities do your students have to make justifications or to use algebra
to make proofs?
Do you have class discussions about why examples are not enough to make a
convincing argument?
As you look at the work of your students, what were the qualities that you valued
in their explanations? What would you have liked to see more of?
What are some examples of problems your students have worked on that required
use of justification? Do you need more resources for finding these types of tasks?
How can you make justification more a part of the normal classroom routine?
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Now look at their work on giving examples in part three.
Were some of your students confused by the definition of consecutive numbers?
What was your evidence?
Did some of your students have trouble with the constraints in part three? Did they
forget to make examples for only even answers? Did they use numbers that
weren’t consecutive?
Now look at your student work for part four. How many of your students put:
6
3
2 odds and an
Other comments about
Other
even
the consecutive numbers
What are the implications for instruction? What would you like your students to
be able to do? How do you set up a classroom that fosters justification and builds
students’ logical thinking skills?
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Looking at Student Work on Odd Sums:
Less than 1% of the students made any attempt to use algebra to explain why every odd number
can be written as the sum of two consecutive whole numbers. Student A is the only example
from the sample set.
Student A
Student B attempts to explain that odd numbers have an “extra one” and makes a justification
based on doubling the even number. The student makes a strong argument in part four about the
structure of numbers and why it will be a multiple of 3. With encouragement, the student might
be able to expand the justification to an algebraic proof.
Student B
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Student B, cont.
Student C also tries to make an argument about the odd having an extra “one” and how that
makes it part of the consecutive pair.
Student C
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Student D gives a procedure for finding the consecutive numbers. In a backwards way, this also
explains why odd numbers can be made from the sum of consecutive numbers.
In part four, the student explains the pattern works for numbers divisible by 3 and then gives
examples to prove the rule.
Student D
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Student E also gives a procedure for how to find the consecutive numbers. Student E is able to
distinguish that the rule in part 4 is really multiples of 6. The student should be encouraged to try
to use the examples to make a more generalizable, or algebraic, case for why the pattern is
multiples of 6. What kinds of prompts might help the student think in more general terms?
Student E
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Student F tries to make an argument from examples, using the idea that each sum is “2” greater.
How might the student have used algebra to make this case stronger? What additional questions
would you like to ask the student?
Student F
Student G just uses the rule about adding odd and even. Rather than completing that argument
by showing how that would make all odd numbers, the student just gives examples that support
the rule. In part four, Student G does not explain how to look at a number and decide if it can be
written as the sum of three consecutive numbers, but instead explains how to find the
consecutive numbers.
Student G
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Student H is just paying attention to odd and evenness of the numbers. The student does not
have the logic of making a justification, but is still trying to identify patterns.
Student H
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Algebra
Course One/Algebra
Task 2
Odd Sums
Student Task
Work with odd, even and consecutive numbers. Make and justify
conjectures about consecutive numbers.
Core Idea 2
Employ forms of mathematical reasoning and proof appropriate to
Mathematical the solution of the problem, including deductive and inductive
Reasoning
reasoning, making and testing conjectures and using
counterexamples and indirect proof.
• Show mathematical reasoning in a variety of ways including
words, numbers, symbols, pictures, charts, graphs, tables,
diagrams, and models.
• Draw reasonable conclusions about a situation being modeled.
Based on teacher observation, this is what algebra students knew and were able to do:
Give examples to fit constraints using consecutive numbers
Understand definitions of odd numbers, even numbers, and consecutive numbers
Knew rules like an odd number plus and even number equals an odd number
Areas of difficulty for algebra students:
Using algebra to make justifications
Giving a process for finding the consecutive numbers rather than making a justification
Noting the pattern of multiples of three in part four, rather than the more specific pattern
of multiples of six
Giving an explanation about the three consecutive numbers in part 4, instead of giving a
rule for how to tell if the answer could be written as the sum of 3 consecutive numbers
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The maximum score available on this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 83%, could find examples of consecutive numbers to make 15 and 99. More than
half the students could also give examples of three consecutive numbers that equaled an even
sum. About one-third of the students could give a rule for telling if an even number could or
could not be written as the sum of 3 consecutive numbers. Almost 12% of the students could
meet all the demands of the task, including explaining why all odd numbers can be written as the
sum of two consecutive numbers. Almost 11% of the students scored no points on this task. 83%
of the students with this score attempted the task.
33
Algebra – 2006
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Odd Sums
Points
0
2
4
6
8
Understandings
Misunderstandings
83% of the students attempted
the task.
Students didn’t understand the term
consecutive numbers and/or made
computation errors.
Students could calculate
In part 3, 10% of the students gave
examples of consecutive
numbers that weren’t consecutive in their
numbers that added to 15 and 99. examples. A few students gave the totals
that would work, without giving the
consecutive numbers that made the total.
Students could give examples of Some students gave examples of odd
consecutive numbers to make 15 numbers. 27% of the students explained
and 99. They could also give 3
how to find the 3 consecutive numbers
examples of even numbers that
rather than how to tell if the answer could
could be made from the sum of
be written as 3 consecutive numbers.
three consecutive numbers.
Students could explain how to
Students struggled with explaining why all
tell if a number could be written odd numbers could be written as the sum
as the sum of three consecutive
of two consecutive numbers. 35% said
numbers.
that an odd plus and even = odd.
Students could give examples
7% gave a process on how to find the
and make justifications about
consecutive numbers. 7% of the students
why odd numbers could be
gave justifications about the extra one in
written as sum of consecutive
odd numbers. Less than 1% attempted to
numbers and identify a rule for
use algebra.
telling if a number could be
written as the sum of three
consecutive numbers.
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Implications for Instruction
Students need more practice with the logic of proof and justification. Students should be able to
reason about the structure of number or use algebra to explain why numerical patterns work.
Students need frequent opportunities to conduct number experiments, organize their data, and
make and justify their conjectures. Logic is learned and honed through practice and discourse.
Students at this level should also be encouraged to apply the algebraic skills they are acquiring to
their justifications.
Looking at problems, such as the many guess my number tricks, is one example where students
might use algebra to justify why the number always works. The 2003 Course One Task: Criss
Cross Numbers or Number Towers, are also good to get students to apply symbolic notation to
making a justification.
Fostering Algebraic Thinking, by Mark Driscoll, is an excellent resource for problems to help
students develop their logical reasoning skills in number theory, generalizing from arithmetic,
and looking at functions.
Ask your coach for a copy of the Problem of the Month, Squirreling It Away. See if your
students can work their way up to level E, making a generalization.
Action Research:
Exploring Number Theory
Ask your students to develop a model for an odd number using picture. How can they make this
model work for any odd number, not just a specific odd number like 3 or 7? Ask them how the
specific number is different from the general model. Now ask them to explore why an odd plus
an odd makes an even or an odd plus an even makes an odd. Finally see if they can transfer this
understanding to symbolic notation.
Have them do an investigation of writing numbers as the sum of consecutive integers. See if
they can find a pattern about numbers that can be written as the sum of 3 consecutive numbers or
4 consecutive numbers. Challenge them to prove their conjectures using algebra.
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Patchwork Quilt
Rubric
The core elements of performance required by this task are:
• recognize and extend a number pattern
• express a rule using algebra
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: 31
section
points
1
Gives a correct explanation such as:
The first black hexagon needs 6 white hexagons and the other five black
hexagons each need five white hexagons. 6 + 5 x 5 = 31
(accept 21 + 10 = 31)
2.
points
Gives correct answer: 13
1
2
1
Gives a correct explanation such as:
The first black hexagon needs 6 white hexagons and the other black
hexagons each need five white hexagons.
66 – 6 = 60
60 ÷ 5 = 12
12 + 1 = 13
1
2
3.
Gives correct answer: W = 5n + 1 or equivalent
2
Partial credit
Gives an expression such as 5n + 1
4.
(1)
Gives correct answer: 386
1
Shows work such as: W = 5 x 77 + 1
1
Total Points
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2
8
38
Patchwork Quilt
Work the task and examine the rubric.
What are the different mathematical demands of the task?
What are the strategies that students could use to solve the task using algebra?
What are the strategies that students could use not involving algebra?
What kinds of experiences help students to bridge between representations:
geometric figures, tables, graphs, pattern recognition, writing a verbal rule, and
writing and using a symbolic rule? How do you help students make those
connections?
How is seeing that a pattern grows by a set amount different from understanding or
developing a rule to solve for any number in the pattern? What does a student
need to understand or pay attention to help develop a rule or function?
Looking at the geometric pattern, describe how the pattern grows? What happens
when two of the patterns connect?
Look at some part of the pattern, say stage four, what are ways that you can think
about to make counting all the white tiles easier? How does the way you “see” the
pattern help you find a rule or generalization that could be used for any stage in the
pattern?
Why is noticing that the pattern grows by 5 each time not enough to make a
general rule?
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Look at student work for part two of the task, working backwards from 66 white
tiles to the stage number or number of black tiles. How many of your students put:
13
12
11
66
Other
What might a student have been thinking about who divided by 6? What is the
student not understanding?
If students divided by 5, the growth rate, what does the remainder represent?
Some students tried to take a value from the table, like two black tiles = 11 white
tiles; so 2 x 6= 12. 11 x 6 = 66 What about the geometry of the pattern makes this
method incorrect? Why doesn’t this rule work?
Now look at student formulas in part three.
How many students gave a correct formula of 5n+1 or 6n – (n-1)? What do
each part of these formulas represent in the diagram?
If a student gave a formula of 4n + (n+1), what parts of the diagram might
he be looking at?
How many of your students tried to use a rule like 6n, because all hexagons
have 6 sides?
5n or n+5 because each time the hexagon grows by 5?
Did any of your students try to introduce a third variable?
How many of your students put both variables on the same side of the
equation?
Now look at the work to part four, how many of your students put:
# Larger
Other
386
385
462
# Smaller
than 500
than 100
What do you think your students don’t understand about variables and how they
are used to represent ideas?
What are the implications for instruction?
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Thinking about the Mathematics of Patchwork Quilt
A major part of the task is for students to understand the idea of a function with a constant: to
recognize what the growth rate is and how the first term includes a constant. Many students at
this grade confuse the properties of proportions and functions. When looking at student work, see
if the student notice’s that the pattern starts at “6” or that the first term is “5+1”. This means that
the student is noticing the constant in the pattern. Also check to see if the student identifies the
growth rate as 5 (each additional tile adds five sides) or as 6 (all hexagons have six sides).
Here are some important properties of proportions and functions.
Proportions
Linear graphs going through origin
Can add or multiply values from
table to extrapolate to further values
Can be expressed as a multiplicative
relation, y=kx
Need to quantify growth rate,
thinking in multiple groups
Functions with Constants
Linear graphs not going through origin – Graph
intersects the y-axis at the value of the constant
(the zero step of the pattern – step one equals the
growth rate plus the constant)
Can’t use table to add or multiply unless you
compensate for the repetition of the constant
Has a multiplicative part but with an extra bit
added on (can be negative or positive number),
y=mx + b
Need to quantify growth rate, thinking in multiple
groups; but also account for the constant being
used only once
The focus of elementary grades is to start seeing and thinking in groups, being able to see equal
size groups as representing multiplication. While many students need to draw and count at these
grade levels, students should start to transition into more efficient strategies, like adding on or
continuing a table. Somewhere in late elementary school, students develop an understanding of
multiplicative relationships. If something is growing in groups of 5 or 6, they should start to use
multiplication of this unit to move to larger numbers. They should be able to use multiplicative
reasoning to extend patterns rather than repeated addition. By middle school, students thinking
should start to move towards finding rules that will work for all numbers; which they will be
trying to express these relationships symbolically. While many students have the logic of the
generalization, their attempts at using the new language of symbols may cause them difficulty.
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Examining the Thinking for Middle School
In the 2006 MAC exams, there is a problem about growing hexagons in grades 3, Toothpick
Houses, grade 5, Hexagons in a Row, and algebra Patchwork Quilt.
Demands of the task:
Demand of task
Visual
DiscriminationDraw the next figure
Find the pattern and
fill in table
Expand pattern
forward
Use inverse
reasoning to find the
stage number
Give a formula
Grade 3
Grade 5
Show the jump from NA
stage 3 to stage 4
Algebra
NA
Stage 1-3 given
Fill in stage 4-6
From 3 to 6
From 6 to 11
From 41
Stage 1-2 given
Fill in 3-4
From 4 to 5
From 5 to 12
From 76
Stage 1-4 given
NA
NA
Find a formula
Use formula to stage
77
NA
From 4 to 6
From 6 to 77
From 66
The demands and challenges do not seem all that different for the algebra students, with the
exception of the need to derive a formula for finding the total number of white tiles, W, given the
number of tiles, n.
In fact many students have not moved from the type of reasoning and often inefficient strategies
used at lower grade levels. However, for other students, they are using new thinking and
problem solving strategies and trying to apply the language of algebra to their thinking. So like
the 3rd and 5th graders, the algebra students are not completely fluent in expressing their ideas
with these new tools. In examining some of their work, the challenge is not the absolute
demands of the task, but the reasoning and expressing the new ideas that makes this task unique
for students at this grade level.
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Looking at Student Work on Patchwork Quilt
Student A, like the students for 3rd and 5th grades, is thinking about how the overlaps effect the
pattern. The student most likely started by continuing the table to solve part two. Then when
pushed to find a formula, the student seems to be using the table to find the relationship between
the independent and dependent variables. The student struggles with the language of algebra and
has to invent a 3rd variable to quantify the idea of doing the arithmetic in two stages. The gaps
are expressed as (n-1). The formula is correct because the student was able to quantify the
variable p in terms of n.
Student A
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Student A, continued
Student B notices the pattern used by many students at lower grade levels. The first term is 6
and all subsequent terms increase by 5, (6+5+5+5…). On the first page of the task the student
notices that the pattern increases by 5 (in the margins are evidence of continuing the table). The
student attempts to quantify this pattern in the algebraic expression in part three. However, if the
expression is examined carefully, the student is not using the W as a variable for white tiles, but
a label for white tiles; so the expression should read 6 white tiles + 5 white tiles + 5 white tiles.
Secondly, the student is not writing an equation for finding any number in the pattern, but an
expression for finding the white tiles for 3 black tiles (3n should read 3 black hexagons). The
student’s thinking is not consistent. Student B loses the “fiveness” in part two, but can use the
pattern to solve for part four.
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Student B
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In 5th and 3rd grade, many students were thinking in units of 5; so 5 x3, to get to the next stage.
This idea of an arithmetic unit was not evident in the work of the algebra students. However, the
algebra students are able to think of the unit as a variable, which is being expressed in the form
5x or 5n. Student C shows the finding of the unit on the chart by finding the constant difference.
Then Student C quantifies the relationship between the independent and dependent variable
using the unit and the constant. Unfortunately Student C does not use inverse reasoning in part
two, treating the white hexagons as black hexagons in the formula.
Student C
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Student C, continued
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In earlier grades, students noted that the first term started with 6 and kept adding 5. Student D
notes this idea and can use it to solve the problems in part 2 and 4. However the student does not
understand how to adequately express that idea in symbolic notation. The first line of the student
work in part 4 could be written as 6+5(n-1) =W. Instead of a formula the student is just
describing the pattern: the first black hexagon has 6 white tiles, each additional black tile adds 5
white tiles. Notice again the letters are being used as labels, not as variables. Although the equal
sign is present, it is not describing a rule or function for solving for any number of black
hexagons.
Student D
Student D, continued
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Student E is still struggling with how to use the language of algebra, not fully grasping the idea of using 2
variables. The student uses w in place of the coefficient 6 in his equation and uses w to represent the
number of white tiles. So w is used to represent two different things. The logic behind the equation is
valid and leads to a correct solution in part 4. The equation the student is trying to write is for the idea of
subtracting out the overlaps.
Student E
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Student F is able to find a rule for the pattern and writes for part one, “You multiply it by 5 then
add 1.” The student can then apply this rule to solve for part four of the task. However the
student struggles with how to account for the constant when working the rule backwards. The
student subtracts the constant after the division instead of before the division and is thus always a
tile short.
Student F
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Student G had a valid logic for finding the number of white tiles, but has not yet learned order of
operations or use of parentheses. The student’s rule would be acceptable if it were written as W=
6 + 5(n-1).
Student G
Some students do not examine all the evidence before writing a rule. Student H comes up with
an expression that only works for the second value in the pattern. The student knows it won’t
work for stage 1, but then doesn’t test it to see if it works for 3 or 4. Notice that the student can
calculate the value of an algebraic expression, but is just using an incorrect expression.
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Student H
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Some students are still using the incorrect strategies or inefficient strategies from previous grade
levels. There has not been a progression in their reasoning. Student I is thinking about all
hexagons having six sides and does not notice the overlap. Even the student’s drawing shows
discrete hexagons, instead of tiling hexagons.
Student I
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Student J tries to use a drawing and counting strategy, but does not apply it correctly. By not
using a continuous drawing the constant (+1) has been included in the count twice. The counting
has distracted the student from noticing that the pattern grows by 5’s. So in part four the student
is dividing by 6 and multiplying by 5, not being able to decide which strategy is correct. Should
the number get larger or smaller?
Student J
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Some students are still relying on the comfortable “continue a table” strategy used at earlier
grades. However, this can lead to calculation errors and is quite cumbersome. See the work of
Student K.
Student K
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Student L just tries to apply the rules of using a table, but in part 4 fails to account for the
constant. The student does not understand functions and writes a recursive rule for the pattern.
Student L
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Student M is not thinking in groups or units. The student just sees the arithmetic of adding 5
every time or subtracting 5. The student has not developed multiplicative thinking.
Student M
In algebra students are transitioning from simple arithmetic to making generalizations, moving
from concrete to abstract. They are not only looking at the patterns, but are trying to think about
how to explain them in symbolic mathematical notation. They struggle with the ideas of variable
(something that can stand for any number in a set) instead of labels and specific solutions. They
need to think about the relationships between independent and dependent variable. Students
confused or unclear about the “idea” of a relationship may still be trying to write expressions
with both variables on the same side of the equation (e.g. w= 4n+(w-4n) or n=(w+5))n) or just
using expressions instead of equations (e.g. w*n or n +5w). While some students are still using
adding on, continuing a table, or drawing and counting, most students are reaching for some sort
of rule. Many have the logic or reasoning of how the rule works, but can’t adequately describe it
symbolically.
Because their thinking and skills were not as consistent throughout the task as students at lower
levels, student strategies are broken down by three major task demands: expanding the pattern to
stage 77, working backwards from white hexagons to a stage number, and writing an equation.
Approximately 42% of the students solved this part by using the rule or doing calculations of 77
x 5 +1. They understood the constant of one and the growth rate of 5. 3% of the students solved
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this portion by thinking about the overlaps ,
(77 x 6) – 76. 3% thought of task as the first
term is 6 and there are 76 additional units of 5 (6+ 76x5). 1% attempted incorrectly to continue
the table to the 77th term. 9% of the students are still thinking about hexagons having six sides
and multiplied 77 x 6. 13% of the students only thought about the growth rate of 5 and ignored
the constant (77 x 5). 5% treated the 77 as if it represented white hexagons. While not common
in work at earlier grade levels, 12% of the algebra students did not attempt this part of the task.
In working backwards from the number of white hexagons to the stage number, 21% could use a
rule or calculations accounting for the constant and growth rate in the form (n-1)/5. 3%
subtracted off the first term, the term with the constant, and then divided by five. They could
also realize that the first term represented the first hexagon and add that answer back to the total
hexagons found by the division.16% extended the table. 2% used a draw and count strategy.
Incorrect strategies included 7% dividing by 6. 3% divided by five but were unable to explain the
remainder of one or the constant. 4% tried to extend the table or add 5’s, but made calculation
errors.
43% of the students could write a correct equation in the form, 5n+1. 3% wrote an correct
equation for the inverse relationship, solving for black hexagons. 8% had expressions with 6 w.
6% wrote some attempt at a recursive expression like w= n=5. 5% did not write equations, but
used expressions involving both variables (e.g. w*n).
Task demands
Extend the pattern to 5 or 6
Extend the pattern to 11,12, or 77
Work backwards from 41, 76,66
Give an equation
%of students meeting the demands
3rd
5th
Algebra
79%
77%
87%
57%
67%
53%
61%
38%
68%
27%
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Algebra
Course One/Algebra
Task 3
Patchwork Quilt
Student Task
Recognize and extend a number pattern for a geometric pattern. Express
a rule using algebra. Use inverse operations to solve a problem.
Core Idea 1
Understand patterns, relations, and functions.
Functions
• Generalize patterns using explicitly defined functions.
and Relations
• Understand relations and functions and select, convert flexibly
among, and use various representations for them.
• Recognize and generate equivalent forms of simple algebraic
expressions and solve linear equations.
Based on teacher observation, this is what algebra students knew and were able to do:
Extend the pattern to five and explain how they got their answer, usually noting the
growth rate of 5.
Work backwards to find the number of black hexagons needed for 66 white hexagons.
Areas of difficulty for algebra students:
Writing a rule or formula
Understanding the difference between a recursive rule and a generalized rule
Understanding that a variable is not the same as a label
Understanding how the first term is different or how the constant effects the progression
of the pattern
Expressing ideas in symbolic notation
Order of operations
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The maximum score available on this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 82%, can extend the pattern to 6 black hexagons and explain how they figured it
out. More than half the students, 61%, could extend the pattern to 6 black hexagons and work
backwards from the number of border tiles to the number of black tiles, giving explanations for
both. Almost half the students, 48%, could do some correct reasoning for extending the pattern
to 77 black hexagons, but they may have made calculation errors. 29% of the students could
meet all the demands of the task including writing an algebraic formula for extending the pattern
to any number of black tiles. 13% of the students scored no points on this task. 89% of the
students with this score attempted the task.
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Patchwork Quilt
Points
0
2
4
6
8
Understandings
Misunderstandings
89% of the students with this
score attempted the task.
Students struggled with extending the
pattern. Many made calculation or
counting errors (32, 33). Some thought
about hexagons having 6 sides (36).
Students could extend the
Students struggled with working
pattern to 6 black tiles and
backwards from 66 white tiles to finding
explain how they figured it out, the number of black tiles. 7% of the
usually noting the growth rate
students solved the problem as if the white
of 5.
tiles were black (331). 13% divided by 6
(11). 4% divided by 5 but didn’t account
for the first tile (12/14).
Students could extend the
Students had difficulty finding a rule that
pattern to 6 black tiles and work would work for all tiles. 12% did not
backwards from 66 to find the
attempt this part of the task. 5% tried to
black tiles.
multiply by 6. Others forgot the constant
or wrote recursive rules. Some students
did not understand the idea of variable and
used the letters as labels.
Students with this score still struggled
with working backwards. While it was the
second thing most students could solve
for, it was also the most difficult for
others.
Students could extend a pattern,
work backwards from an
answer to the stage number, and
use symbolic notation to write a
formula that included a growth
rate and a constant.
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Implications for Instruction
Students at this mathematical level would benefit from working a variety of problems in context
to help them quantify ideas in symbolic notation and gain a clearer understanding of the purpose
and meaning of variables. Students need to see the usefulness of variables in solving problems.
Working a mechanical procedure for solving an equation is quite a different skill from
expressing an idea in the language of algebra.
Context of pattern problems allows students to search for generalizations and use the context to
justify why the generalization holds true for all cases. One of the big mathematical ideas and
purpose for algebra is to be able to make and prove generalizations.
Students at this grade level should be doing detailed investigations to explore proportions and
functions to understand the properties of both. How are they the same? How are they different?
Context makes these similarities and differences more apparent.
Consider a proportional situation, such as t-shirts cost $5 apiece. The table for this proportional
relationship (5x) might look like:
1
2
3
4
5
6
5
10
15
20
25
30
Then it would be quite valid mathematically to take the second term (10) and multiply it by 3 to
get 30 or add the 4th term + the 4th term + 3rd term to equal the 11th term (20+20+15 = 55 or 5
(11)= 55). However, this reasoning is not valid for functions with a constant. Students try to
make generalizations such as these about patterns from work with proportion situations and don’t
realize that the procedures will not work for all patterns.
However for a pattern like Patchwork Quilt, or the 5th grade task: Toothpick Houses, both of
which grow by 5’s, the rules of proportions don’t work. See the work of student E and F.
Student E
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Student F
Exploring the properties of proportions and functions with constants should also include a
comparison of their graphs. Students should be asked what the constant represents in the context
of the problem to help them understand why it can’t be repeated.
Ideas for Action Research
Exploring Proportions and Functions with Constants
Design a lesson to explore the differences between proportions and functions with constants.
See if students can see where the constant is within the context and why that element is not
repeated in future iterations of the pattern. Have them graph the equations and see if they can
identify the constant within the graph. Here are three problems that might help you with the
lesson planning. The lesson might include have students make a poster to compare the three
problems.
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Poster for all problems:
Make a table. How could the table help you solve the task?
Make a graph. How could the graph help you solve the graph?
Describe the pattern in words. What stays the same? What changes? How is the
pattern growing? What does this make you pay attention to?
Initially, did you use one of these three strategies or did you do something
different?
Questions for discussion might include: What is the same and what is different
about the tables? About the graphs? Where is the “threeness” in each problem?
Where is the +1 in each problem? Where do these show up in each representation?
How are the problems the same or different?
Problem 1:
Addworms
How long will the Addworm be when it is 5 years old?_____________
How long will the Addworm be when it is 10 years old?____________
Explain or show how you figured it out.
How long will the Addworm be when it is 45 years old?___________
Explain or show how you figured it out.
Can you write a rule for finding the length of an Addworm of any age?
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Problem 2: Toothpick Towers
How many toothpicks would it take to build shape 5?____________
How many toothpicks would it take to build shape 10?___________
Explain or show how you figured it out.
How many toothpicks would it take to build shape 45?_____________
Explain or show how you figured it out.
Could you write a rule for finding the number of toothpicks needed to build any shape number?
Problem 3: Growing T’s
How many tiles would it take to make the 5th T?____________
How many tiles would it take to make the 10th T?___________
Explain how you figured it out.
For the 10th T, how many tiles would go across the top? How many tiles would make the stem?
How many tiles would it take to make the 45th T?_______________
Explain how you figured it out.
For the 10th T, how many tiles would go across the top? How many tiles would make the stem?
Could you write a rule to find the total number of tiles for any size T? For finding the number of
tiles across the top of the T? For finding the number of tiles needed to make the stem?
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Printing Tickets
Rubric
The core elements of performance required by this task are:
• compare price plans using graphs and formulae
Based on these, credit for specific aspects of performance should be assigned as follows
points
1.
Gives correct formula such as: C = 10 + t / 25
1
2.
Draws a correct straight line from: (0, 10)
to (400, 26)
2
Gives correct answers: C = 20
t = 250
1
1
3.
1
2
Shows correct work such as:
2t ÷ 25 = 10 + t ÷ 25
2t = 250 + t
C = 2 x 250 ÷ 25
4.
section
points
1
1
Gives a correct explanation such as:
If Susie buys less than 250 tickets, Sure Print will be cheaper,
and if she buys more than 250 tickets, Best Print will be cheaper.
4
1
1
Total Points
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9
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Printing Tickets
Work the task and examine the rubric.
What are the mathematical demands of the task?
Look at student work on writing a formula. How many of your students put:
(t/25) +1
(t+10)/25 (10t+1)/25
1t/25
11t/25
10t/25
25t/10
Other
What big mathematical ideas are confusing for students?
Which errors are related to order of operation or understanding symbolic notation?
Which errors are related to not understanding the constant?
Which errors are related to not understanding the rate of change?
How often do your students work with making their own graphs? Do they make their own graphs
from scratch or is most of the work done on graphing calculators? Do students have the habits of
mind of making a table of values before completing a graph?
Look at student work on graphs.
Did their graphs match the values that would have fit their equations?
Yes________________________ No_____________________________________
How many understood that the graph should intercept the cost axis at $10 because of the
constant or fixed cost?______________________________
Look at student work on part three:
How many of your students set the two equations equal to each other and solved for the
unknowns?
How many of your students used substitution of values most likely obtained from reading
their graph?
How many of your students picked the original values C=$2, t=25 or C=2t/25?
Did your students think about more than 250 tickets and less than 250 tickets in part four? Did they
think about large amounts and small amounts, but not quantify those values?
Did their answers match their graphs?
Did students think Sure Print would always be the best because you wouldn’t have to pay the set up
costs?
What surprised you about student work? What are the implications for instruction?
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Looking at Student Work on Printing Tickets
Student A is able to write an equation to represent the costs of using Best Print and graph the
equation. The student knew to set the two equations as equal to find the number of tickets when the
costs would be the same. The student could quantify under what conditions it was better to use each
print shop.
Student A
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Student B is also able to use algebra to solve the task. The student uses the idea of two equations
with one unknown and solves the second equation for t. Then the student substitutes this solution
for t into the first equation.
Student B
Student C is able to write an equation and graph it on page one of the task. Like many students, C
seems to use the graph to find the point where the two values are the same and substitutes those
values into the two original equations. Student C can’t use the graph or information in part three to
choose when to use each Print Shop. The student thinks that Susie should never choose Best Print
because the $10 set up fee is too high.
Student C
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Student D is able to write an equation for Best Print and make its graph. The student seems to make
notations at the bottom of the graph to help compare Sure Print with the new graph of Best Print.
However the student isn’t able to use this information to answer part three. The student does use the
graph correctly to reason out part foour, determining the conditions for using each print shop.
Student D
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Student E is also able to write the equation and draw the graph. However the student uses only the
costs of buying 25 tickets to compare each shop, rather than considering the values for buying a
whole range of tickets and makes an incorrect conclusion.
Student E
Student F is unclear how to use symbolic notation to calculate the values for Best Print. However,
the student seems to know the process and use it to make a correct graph. Notice all the points on
the line, which seem to indicate calculated values. The student is able to look at the graph and see
that costs are the same at 250 tickets. However when the student substitutes the values into the two
equations, reasoning seems to shut down as the student finds values. The student takes the answers
to the two equations after substitution and then divides. What might the student be thinking? What
would you like to ask the student?
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Student F
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Student G writes a common incorrect equation. The graph does not match the results of calculations
with that equation. Student G does not seem to understand the rate being used in this problem. The
student sees the 25 in the two equations as the variable for number of tickets, not part of the rate 2
for 25 or 1 for 25. See the work in part three. The student seems to understand the general context
of the problem and can reason (without quantity) when to use each print shop.
Student G
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Algebra
Course One/Algebra
Task 4
Printing Tickets
Student Task
Compare price plans using graphs and formulae. Use inequalities in a
practical context of buying tickets.
Core Idea 3
Represent and analyze mathematical situations and structures
Alg. Properties using algebra.
&
• Write equivalent forms of equations, inequalities and systems
Representations
of equations and solve them
• Use symbolic algebra to represent and explain mathematical
relationships
• Judge the meaning, utility, and reasonableness of results of
symbolic manipulations
Based on teacher observation, this is what algebra students knew and were able to do:
Write an equation for Best Print
Draw a graph to match their equation
Interpreting graphs of two equations to determine best buy under different conditions
Areas of difficulty for algebra students:
Understand how to use symbolic notation to represent a context
Find a table of values before drawing a graph
Using algebra to solve for 2 equations with 2 unknowns
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The maximum score available on this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
More than half the students, 60%, could write an equation to represent the cost of buying tickets at
Best Print. Almost half the students, 40%, could also graph the cost of Best Print. Some students,
could find when it was cheaper to use Best Buy or Sure Print. 8.5% of the students could meet all
the demands of the task including using algebra to find the point where the costs for Best Buy and
Sure Print are the same. Almost 40% of the students scored no points on this task. 90% of the
students with this score attempted the task.
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Because of the number of students scoring zero, their thinking is documented below:
Common Equation Errors:
(t+10)/25
(10t +1)/25
1t/25
11t/25
10t/25
25t/10
6%
3%
14%
16%
11%
3%
Graphing Errors:
58% of the graphs did not match the equation the student had written for part 1.
21% of the graphs had lines parallel to Sure Print
43% of the graphs for Best Print went through the origin (0,0) which would only be true if there
were no constant ($10 set up)
Finding values for C=___ and t+______:
No answer – 41%
Picking 25 tickets with some other value: 30%
Picking the best print shop:
Sure Print and matches their graph: 24%
Sure Print, doesn’t match their graph: 6%
Sure Print, some other reason: 11%
Best Print, matches their graph: 6%
Best Print, doesn’t match their graph: 8%
Best Print, other reasons: 8%
Printing Tickets – for all students
Points
Understandings
Misunderstandings
90% of the students with this
40% of all students scored no point on this
0
score attempted the task.
task. See analysis of their work above.
Students
could
write
an
Students did not understand how to use the
1
equation to represent the cost
constant of $10. Some treated it as a variable,
of tickets using Best Print.
11t over 25 or 10t/25 or (10t +1)/ 25. Some
students struggled with order of operations
(t+10)/25.
Students could write and graph Many students did not make graphs that
3
an equation for Best Print.
matched values that could be obtained from
their equations. There was no evidence of
making a table of values before making their
graphs.
No clear pattern.
4
Students could write and graph Students could not use algebra to solve for
7
the equation. They could
when the costs were the same. Most students
determine when the costs were used the information from their graphs and
the same for both companies.
substituted the values into the two equations.
They could explain which
company to use in different
situations.
Students could meet all the
9
demands of the task including
using algebra to find when the
costs were equal.
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Implications for Instruction
Students at this level need more opportunities to use algebra in a practical situation. Students should
have practice making a table of values to help them graph equations. They should also understand
how a constant effects the graph and be able to use the formula to think about slope.
Some students at this level are still struggling with understanding the meaning of
variables. They see the letters or symbols as standing for labels. Others think that an
equation is only for finding one specific value. They don’t understand that the letter
represents a quantity that can vary or change. Students need more experience with
solving problems in context that promote discussion about how the variable may
change and why. They need to connect the equation to a wide range of possibilities,
to a representation of a more global picture of a situation. These nuances do not come
through practice with just symbolic manipulation.
A few students struggle with the basic algebraic notation around order operations,
combining algebraic fractions and whole numbers, and solving equations with
divisors or fractional parts.
There are many situations where it is important to find the breakeven point or place
where two functions intersect. Students should be familiar with these types of
situations and be comfortable setting the two equations to equal each other. These
problems might include choosing the best rate for a gym membership, picking a cell
phone plan, or ways to price a store item with different costs having different volumes
of sales.
(See MAC tasks – 2003 6th Grade: Gym, 2005 8th:Picking Apples)
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Algebra – 2006
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Rubric
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Graphs
The core elements of performance required by this task are:
• relate given line graphs to their equations
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives 7 correct labels (see below)
section
points
5
Partial credit
6 correct labels
5 correct labels
4 or 3 correct labels
2 or 1 correct label(s)
2.
points
(4)
(3)
(2)
(1)
Gives correct answer: (6,6)
5
1
1
3.
Gives any correct line, for example, y = 6, x = 3, y = x + 3, y = 9 – x, etc.
1
1
7
Total Points
B
G
C
J
F
E
A
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Graphs
Work the task and examine the rubric.
What are the key mathematics the task is trying to assess?
Look at student work on labeling the graph. How many of your students
labeled these parts incorrectly:
B
G
C
E
F
J
A
What do you think your students had difficulty understanding?
Now look at your equations for a straight line going through the point (3, 6).
Make list of student equations, make tallies for those that are repeated:
Correct
Incorrect
What do these equations suggest that your students don’t understand about graphing and
equations? Are there common threads to the errors?
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Looking at Student Thinking on Graphs
For the blank on the graph with the expected answer of B, the line x=0:
27% put A, the line y=0. Students mixed up the two axes.
5% put G, the line x + y =9, even though the bubble is pointing at 10 on the yaxis.
For the blank on the graph with the expected answer of G, the line x + y = 9:
6% put I, the line y = x - 6
5% put D, the line y=6
5% put K, the solution of the simultaneous equations x + y = 9 and y = 2 x
7% put H, the line y = x + 6
5% also picked C, J, and F
For the blank on the graph with the expected answer of C, the line x = 6:
13% put D, the line y = 6
4% put the answer J, the solution of the simultaneous equations x + y = 9 and y =
1/2x
For the blank on the graph with the expected answer of E, the origin:
10% put A, y= 0
4% put K, the solution of the simultaneous equations x + y = 9 and y = 2x
3% put J, 3% put K, 3% put B, 3% put I
For the blank on the graph with the expected answer of F, the line y = 1/2 x:
6% put I, the line y = x – 6
4%% put K, the solution of the simultaneous equations x + y = 9 and y + 2x
4% put J, the solution of the simultaneous equations x + y = 9 and y = 1/2 x
3% each picked H, E, C, A, B, and H
For the blank on the graph with the expected answer of J, the solution of the simultaneous
equations x + y = 9 and y = 1/2x:
12% put C, the line x + 6
10% put G, the line x + y = 9
9% put K, the solution of simultaneous equations s + y = 9 and y = 2x
6% picked B, the line x = 0
6% picked H, the line y = x + 6
5% picked F, the line y = 1/2 x
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For the blank on the graph with the expected answer of A, the line y = 0:
30% put B, the line x – 0
5% put K, the simultaneous equations x + y = 9 and y = 2x
For part two, which point is on the line y = 6 and on the line x = 6, almost 30% of the
students tried to give one of the letter options from part one rather than giving the
coordinates (6,6). Students giving coordinates picked 6 as only one of the two
coordinates.
When asked to give an equation for a straight line going through the point (3,6), many
students tried to use the idea of y = mx +b:
Some examples of these equations:
y=mx + b
y=3m+b
y=bx +b
y = 3x + 6
Other choices show a confusion about equations; such as trying to find a specific
solution, not understanding exponents, having solutions that would not be true for the
values given if substituted into the equation:
x=3+6
x3 + y6 = xy18
y = x+9
9x +3y
y=x+6
3x + 9y
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Algebra
Course One/Algebra
Task 5
Graphs
Student Task
Relate line graphs to their equations.
Core Idea 3 Alg. Represent and analyze mathematical situations and structures
Properties &
using algebraic symbols.
Representations
• Understand the meaning of equivalent forms of expressions,
equations, inequalities, or relations
• Write equivalent forms of equations, inequalities, and systems
of equations and solve them
Core Idea 1
• Analyze functions of one variable by investigating local and
Functions and
global behavior, including slopes as rates of change, intercepts
Relations
and zeros.
Based on teacher observation, this is what algebra students knew and were able to do:
Students could identify the origin
Students could identify the equation x + y = 9
Students could identify the equation y = 1/2 x
Students could give the coordinates for the intersection of y=6 and x = 6
Areas of difficulty for algebra students:
Confusing the order of x an y in a coordinate pair
Recognizing the solutions for simultaneous equations
Writing an equation for a line through a given point
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The maximum score available on this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 3 points.
Most students, 85%, could identify the origin and the line x + y = 9. About half the
students could also identify the line y = 1/2 x. Some students, about 35%, could also
identify the lines x= 0 and
y = 0 and x + 6. 14% of the students could meet all the demands of the task including
identifying the simultaneous solution for x + y =9, giving the coordinates for the
intersection of y=6 and x=6, and writing the equation for a line going through (3,6).
Almost 15 % of the students scored no points on this task. 57% of the students with this
score attempted the task.
87
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Graphs
Points
0
1
3
6
7
Understandings
Misunderstandings
Only 57% of the students with
this score attempted this task.
Students could identify the
origin and x + y = 9.
Students confused the order of x and y in a
coordinated pair or the x and y axis.
Students struggled with the solution of a
simultaneous equation and the line y =
1/2x.
Students could identify the
Students had difficulty finding the
origin and x + y = 9, and either
intersection of the lines y=6 and x =6.
identify the solution of a
Many students tried to pick from the
simultaneous equation or the
options in part 1 of the task, rather than
line y = 1/2x.
trying to graph and find the point for
themselves.
Students could also give the
Students struggled with the solution of
equation for a line going through simultaneous equations and the intersection
the points (3,6).
of two lines.
Students could meet all the
demands of the task by matching
equations or points with their
representation on a graph,
finding the intersection of two
lines, and writing an equation
for a line passing through a
given point.
Implications for Instruction
Students need more work with coordinate graphs and plotting points. Students at this
level are still confusing x an y; for example choosing the line y= 0 foe x = 0. They may
need more experience with making a table of values and plotting points in order to verify
the graphs of lines. They didn’t seem to use substitution of values to verify if the ordered
pair would match their equation. They had difficulty finding the intersection of the
equations on a graph.
Is there some piece of conceptual understanding that gets missed if students don’t
get enough hands on experience before jumping to graphing calculators?
How can you interview students or set up an experiment to find out what they
understand about the connections between equations and graphs?
How does understanding the intersection of two equations relate back to Printing
Tickets?
When teaching finding the solution to simultaneous equations is enough emphasis
put on the meaning of the intersection or the connection between the solution and
the graphs of the equations? How does context help to illuminate this
significance?
How does understanding the graphs of lines help students to develop a better
understanding of the idea of a variable, something that can change or vary? What
contexts help to support students’ development of this big mathematical idea?
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Ideas for Action Research:
Using Cognitive Dissonance:
Cognitive dissonance is based on the idea of presenting conflicting views or ideas and
then grappling with the issues around making sense of which was is true. This is a
technique to develop the logical reasoning of students and primarily used to confront
head-on misconceptions to help students see the errors in their thinking. Based on the
issues that came up on this problem, it might be nice to use the graphic from page one
without the choices from page 2. Have students look at the bubble for the expected
answer G.
Problem: Fred thinks the answer is the line x + y = 9. Franny said is incorrect, because
equations for graphs always needed to start with y. The answer is the equation y = x + y.
Francisco says that Franny is partially correct. Equations for graphs always start with y,
but she mixed up the operations. The equation should be y = x – 6.
Have students discuss in pairs
a) What could each student have been thinking about? What are they looking at?
b) Who do you think is correct? How could you convince your classmates?
Hopefully the discussion will generate ideas about the difference between looking at just
one point on a line versus looking at several points on a line. The discussion could also
bring up the idea that one of the students is confusing the x and y coordinates. In
generating their proofs, students might use a table of values to show that several points on
the line match one of the equations. See what your students bring to the discussion.
Now give your students a coordinate pair, for example (3, 6). After students have finished
their work, pick 3 or 4 of their examples (don’t be afraid of using some that are incorrect,
but at least 2 should be different and correct) to put on the board. Ask the question could
these all be right? How could there be more than one answer? Which one do you think
is correct?
Again have them go back to their pairs to discuss the different solutions. Then, have
students present their ideas to the class. See if they use the ideas about a table of values
from the previous discussion or substitution to help convince their classmates. What is
the logic they are using in their justifications?
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