Shelves
This problem gives you the chance to:
• solve problems in a spatial context
• identify and distinguish the four point graphs related to this situation
Pete is making a bookcase for his books and other stuff.
He already has plenty of bricks and can get planks of wood for $2.50 each.
Each plank of wood measures 1 inch by 9 inches by 48 inches. Each brick measures 3 inches by 4.5
inches by 9 inches.
For each shelf, Pete will put three bricks at each end then put a plank of wood on top. The diagram
shows three shelves.
3 inches
1. Pete wants five shelves in his bookcase.
a. How many planks of wood does he need?
b. How many bricks does he need?
c. How high will the shelves be?
d. How much will the bookcase cost?
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Shelves Test 8
The diagram below shows graphs with the following descriptions:
Description One: The cost of the bookcase against the number of shelves.
Description Two: The number of bricks against the number of shelves.
Description Three: The height of the bookcase against the number of shelves.
Description Four: The width of the bookcase against the number of shelves.
The equations of the graphs are
y = 48,
y = 10x,
y = 6x,
y = 2.5x
100
90
80
70
60
A
B
C
D
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
Number of shelves
2. Complete this table to match each graph with its description and its equation.
Graph letter
A
Description number
Equation
B
C
D
8
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Shelves Test 8
Task 4: Shelves
Rubric
The core elements of performance required by this task are:
solve problems in a spatial context
identify and distinguish the four point graphs related to this situation
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: 5
1
Gives correct answer: 30
1ft
Gives correct answer: 50 inches
1ft
Gives correct answer: $12.50
1ft
4
2.
Four points for eight correct answers.
4
Partial credit
7 or 6 correct 3 points
5 or 4 correct 2 points
3 or 2 correct 1 point
(3)
(2)
(1)
Total Points
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4
8
Shelves Test 8
Shelves
Work the task. What are the big mathematical ideas being assessed?
Students could interpret part 1 in a couple of different ways – how much is needed
altogether or how much more is needed (the task explains that they are looking at a
diagram not at an actual bookcase). Look at student work for interpretation and
consistency.
• How many of your students interpreted the task as written, total amount of
materials needed?
• How many of your students interpreted the diagram as a picture of an already
existing bookshelf (needing 2 more planks, 12 more bricks, height 20 in., and a
cost of $5)?
Now look at other answers for planks:
• How many students put 6_______________? How many put 3____________?
• How many put 10?____________________ Other answers________________
Look at the number of bricks: How many only considered bricks for one side of the book
shelf- 15____________________ 6_____________________
How many made other errors: 18___________ 24_____________other_____________?
Can you figure out how or why students made some of these errors?
Look at height of the bookshelves. How many of your students put:
50
45
9
20
15
54
Other
Why was this difficult for students?
How often do students in your class have opportunities to interpret diagrams?
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Now look at work on matching graphs to equations and descriptions. How do your
students compare with the sample population:
Graph
Description Number
Equations
Letter Correct
Other responses
Correct
Other response
Answer
answer
3
1
2
4
y=10x
y=6x
y=48 y=2.5x
A
12%
14%
7.5%
7.5%
5%
5%
2
1
9%
3
11%
4
11%
y=6x
y=10x
7%
y=48
2%
y=2.5x
9%
4
1
7.5%
2
5%
3
13%
y=48
y=10x
5%
y=6x
4%
y=2.5x
6%
1
2
10%
3
7.5%
4
9%
y=2.5x
y=10x
6%
y=6x
4%
y=48
9%
B
C
D
Did any of your students try to write their own equations?
Why do you think it was easier for students to match the graph with the equation than
with the description?
How often do students work with variables in context?
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Looking at Student Work on Shelves
Student A draws the bottom shelf to get a sense of dimensionality. Calculations are
shown in the margin. Notice how the descriptions are labeled on the side of the graph to
make sense of each line.
Student A
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Student A, part 2
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Student B is able to think about the situation including adding in the height for the
boards. The student forgets to think about the size of the brick, assigning each a unit
value of one, instead of the 3 inches. The student received full marks on the graph.
Student B
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Student C adds an extra board to the bottom of the bookshelves. In finding height, the
student finds the height of one section of bricks rather that the height of all the shelves.
Like Student A, Student C labels the sides of the graph with the descriptions.
Student C
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Student D is one of the few students to draw a diagram of the whole bookcase. The
student thinks about adding on the two boards and the bricks needed for 2 more boards.
The student may have lost the scale for bricks and found the height of the twelve
additional bricks. However for cost the student finds the correct cost for all 5 shelves. So
there is inconsistency in the chain of reasoning. How often do students get practice
working on tasks where they have to use information for a series of steps? Where they
need to continue to visualize and to quantify the same situation? Student D draws lines
on the graph, but leaves the table blank.
Student D
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Student D, part 2
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Student E adds an extra board to the bottom of the bookshelves when figuring height but
not when finding number of planks. The thinking is inconsistent. The student assigns the
cost of the boards to the bricks. The student is able to match all the equations to the
graph, but incorrectly marks 3 of the 4 descriptors.
Student E
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Student F also struggles with inconsistency. Sometimes the student thinks about the
entire bookshelf and can describe it. Then the student only thinks about the additional
bit. Do students get enough opportunity to make their own diagrams to help them think
about the entire situation? What other strategies might have helped students to think
about this situation?
Student F
Student G originally thinks about the entire bookshelf, but answers the other 3 questions
for only the part modeled in the diagram. However the student has a full grasp of the
algebraic representations of the situation.
Student G
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Student G, part 2
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Student H finds the height for 30 bricks, but only the 3 pictured boards instead of the 5
needed. The student struggles with the algebraic representations.
Student H
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Student I is not able to make sense of the first part of the task. The student even has
trouble multiplying with decimals in find the cost of the planks. However the student is
able to correctly match each graph with the correct equation. What do you think the
student understands? What is your evidence?
Student I
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8th Grade
Student Task
Core Idea 3
Algebra and
Function
Task 4
Shelves
Solve problems in a spatial context. Identify and distinguish the pour
point graphs related to this situation.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity and
change.
• Identify functions as linear or nonlinear, and contrast their
properties from tables, graphs, or equations.
• Explore relationships between symbolic expressions and graphs
of lines, paying particular attention to the meaning of intercept
and slope.
Based on teacher observation, this is what eighth graders know and are able to do:
• Find the cost of the bookshelves by multiplying decimals
• Find the number of planks needed for the bookshelf
• Matching equations to graphs
Areas of difficulty for eighth graders:
• Finding the height of the bookshelves by including both the height of the bricks
and the height of bricks
• Keeping a consistent picture of the situation in their head. Students would think
about the total picture for one part, the needed amount for another, and then often
calculate what was in the diagram for another. They don’t seem to have a strategy
for thinking about the information in an organized way or doing an extended
chain of reasoning. Few students made a diagram to think about the entire
structure.
• Matching descriptions to either a graph or an equation.
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The maximum score available on this task is 8 points.
The minimum score needed for a level 3 response, meeting standards, is 4 points.
Most students, 83%, could either find the cost of the boards or match two or three items
on the table. Less than half the students, 44%, could find the boards, bricks, and cost and
match two or 3 items on the table. About 12% could match all the representations
between graph, equations and descriptors and find boards, bricks, and cost of the
bookshelf. Almost 5% could meet all the demands of the task, including finding the
height of the bookshelves. Almost 17% of the students scored no points on this task. 75%
of the students with this score attempted the task.
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Shelves
Points
0
1
4
6
7
8
Understandings
75% of the students with this
score attempted the task.
Students could either match 2
or 3 items in the table or find
the cost of the boards.
Students could match 2 or 3
items on the table and find the
number of planks and the cost.
Students could find all the
measures on part 1(planks,
bricks, height, cost) and match
the equations to the graph.
Students could find planks,
bricks, and cost and match all
the parts of the table.
Misunderstandings
Students had difficulty matching equations
to graphs. See table in the questions for
reflection.
Common errors for cost of boards included:
$7.50, $10.50, $5.00.
Common errors for number of planks: 2
(19%), 6 (10%), 3 (5%). Common errors for
number of bricks: 15 (13%),12 (14%), 18
(7.5%), 6 (5%)
Students struggled matching the
descriptions to the graph. See table in the
questions for reflection.
The struggled with the height of the
bookcase. Common errors: 45 (18%), 9
(12%), 54 (5%) 15 (5%), 20 (4%) and 24
(4%)
Students could move from a
partially completed diagram to
reason about all the measures
in a completed bookshelf
(planks, bricks, height, cost).
Students could match a graph
to its equation and a
description of the context
being represented.
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Implications for Instruction
Students need more opportunity with reading and interpreting diagrams. Many students
are treating the diagram as a picture of a bookcase rather than a schematic for how to
build it. Students did not think about drawing the complete diagram as a thinking tool.
Adding lines or marks to diagrams becomes critical to solving many geometric problems
as well. This is a skill that should be encouraged. Students need many experiences
working tasks with diagrams or where diagrams or models are tools for finding the
solution.
Many students did not include the board dimension in calculating the height of the
bookcase. Students also forgot to include enough bricks to hold up both sides of the
bookshelves. They don’t have enough experience thinking about what parts of a diagram
are relevant.
Students seemed to understand how to match equations to points on a graph but had
difficulty with matching descriptions to equations or points on a graph. Students need to
have more experiences working with algebra in a context to see it as a tool for sensemaking and as a way of representing practical relationships.
Ideas for Action Research – Investigation Professional Research
How do we help students learn to use tools like building diagrams or relating symbolic
notation to descriptions of situations? What needs to happen in students’ brains to
connect between different representations, such as graphing, equations, and diagrams?
We often talk about what needs to be taught or how to teach, but do not spend much time
thinking about how students learn.
Plan a department meeting to focus on the use and learning of representations. Ask
colleagues to bring articles or materials to share and discuss. A good source to start for
looking at articles is the NCTM Yearbook, The Roles of Representation in School
Mathematics.
Article 1 looks at some of the theoretical issues on building internal and external systems
of representation. “Even a high level of (skillfully performing arithmetic and algebraic
computations) does not imply an understanding of mathematical meanings, the
recognition of structures, or the ability to interpret the results. “
Article 7 looks at how students build diagram literacy. What are the stages a student
needs to go through to learn to use and to make diagrams?
Article 14 looks at promoting multiple representations in algebra.
Article 19 explains some of the research used by the Freudenthal Institute in developing
their Realistic Mathematics Education and how they make use of representation,
including instruction on equations from research on developing algebra units for middle
grades students in Mathematics in Context.
Read and discuss the articles. What are the implications for how you teach? For the
kinds of tasks you choose? For the kinds of questions you ask? Combined with the
reading and examining student work, can you pick one goal that you want to investigate
in your classroom? What might be some of the first steps for conducting an investigation
about the role of representations?
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