Ratios and Proportional Reasoning
Assignment #4
Erica Sautter
Marist College
7th Grade Ratios & Proportional Reasoning
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Abstract
One of a teacher’s many responsibilities is to continually assess each of their student’s learning
throughout each topic or unit studied in class. Using assessments such as traditional tests and
quizzes alone is not ideal as they tend to cover only one small portion of the content therefore not
truly covering each student’s fullest ability. To fill this void, teacher’s now include performance
based assessments in the classroom. This type of assessment allows the teacher to assess multiple
objectives at various levels of Bloom’s taxonomy at once, something traditional assessments
cannot. This paper will follow a seventh grade mathematics teacher through the process of
creating a performance based assessment and a rubric to be used with the assessment for a unit
on ratios and proportional reasoning based on the learning objectives shown in a test blue print
created for the unit. The teacher will then reflect on why this particular performance based
assessment is appropriate for this particular class.
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In today’s world of education teachers are being asked to create new ways to assess their
student’s learning. Traditional assessments such as tests and quizzes are no longer filling all of
the gaps. Therefore, in order to fill these voids teachers are being asked to incorporate
performance based assessments into their curriculum. Performance based assessments are
beneficial as they assess students at much higher levels of Bloom’s taxonomy than traditional
assessments do as well as cover several objectives at one time. When it comes to administering
traditional assessments students may know what the correct answer is; however they may not
understand why that is the correct answer or be able to explain to a classmate or the teacher how
they determined their answer (Park & Lane, 1996). Performance based assessment takes the
guessing of how an answer was achieved out of the equation as students are expected to explain
how they arrived at their answer.
Performance Based Assessment
Performance based assessments provide teachers with resourceful information about their
students’ thinking and reasoning as they work through problems to reach solutions and describe
in detail explanations for the solutions they reached (Park & Lane, 1996). Performance based
assessment works by allowing teachers to observe students as they demonstrate their
understanding of multiple skills and knowledge of the subject generally during a hands-on
project or task (Hancock, 2007).
Applying performance based assessments to the classroom brings along many advantages
to both the student and the teacher. One advantage of such an assessment is that it allows the
teacher to clarify why students are expected to reach these learning targets (Hancock, 2007).
Many times giving students a performance based assessment allows the teacher to connect
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learning to a real life situation. Performance based assessments are also beneficial because they
allow teachers to assess each student’s thought process and the extent of their knowledge of the
skills and knowledge being assessed all while they are completing the assessment (Hancock,
2007). This is vital to the teacher as they can draw conclusions as to how students would
approach such a problem if they were to face a similar scenario in the real world during their
adult life. This is important as one of the goals of school is to prepare our students for the real
world and the fact is there are no paper and pencil exams in the real world, but there are
performance based assessments.
When it comes to performance based assessments the advantages usually outweigh the
limitations and disadvantages, however that does not mean they do not exist. Performance based
assessments have three primary limitations including: reliability, sampling, and time (McMillan,
2011). One of the biggest disadvantages and limitations of performance based assessments is
time. It takes teachers ample amounts of time to create good performance based assessments
based upon the standards and objectives being covered in each unit. Then it takes a lot of time
for students to actually complete the assessment which then teachers have to take the time to sit
down and grade (McMillan, 2011). Another limitation of performance based assessments is
reliability. With performance based assessments there is no one answer that is expected for each
problem. This allows room for inconsistencies in performance levels of students which can
produce inaccurate conclusions over time (McMillan, 2011). This reliability can be lessened with
the use of rubrics, which will be discussed a little later, as they provide a guide as to what the
teacher is expecting students to show through the assessment. Sampling is another weakness of
performance based assessments as there are only a few samples teachers will be able to use with
classes due to the time they take to create (McMillan, 2011). With this type of assessment
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samples do not provide adequate information to students as these are previous students’ own
interpretation of the assessment situation.
The greatest disadvantage of performance based assessments would be the time teachers
have to take to create them as well as a rubric to use to grade them, then the time students have to
spend actually performing the task and the time that teachers have to spend grading and
providing feedback. Although this is the greatest disadvantage with this assessment this is
definitely outweighed by all that a performance based assessment has to offer our students as
they reveal each students understanding of the content. It is important to remember as we assess
these assessments the length of the response is not what is important, it’s the extent of the
content in the response that matters most (Park & Lane, 1996).
Rubrics
Rubrics can be used in any content area and are especially useful when it comes to
performance based assessments as they allow students to know exactly how they will be graded
and that all grading will be done objectively by the teacher (Gezie, Khaja, Chang, Adamek, &
Johnsen, 2012). Rubrics take out the idea that an assignment will be graded subjectively based
on who the student is and how the teacher feels about the student by providing guidelines for
grading. Rubrics are “guidelines that clearly articulate performance expectations and proficiency
levels” (Gezie et al., 2012, p.422). They are generally handed out to students as they begin the
assessment allowing students to know what qualities the teacher will be looking for in their
work.
Rubrics can be used in any content area for many different purposes. They can be used
for individual and group projects, oral presentations, practicing favorable behaviors, and any
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other type of performance based assessment (Gezie et al, 2012). Rubrics aid in enhancing student
learning and their end product by addressing the key elements of the learning goal for the
assignment (Gezie et al., 2012).
There are two types of rubrics that teachers can use: holistic and analytic. With holistic
rubrics the teacher gives the assessment an overall score where components are not scored
individually (Gezie et al., 2012). In analytic rubrics the teacher separates the rubric into the
different components of the overall product and each component is given an individual score.
Then each of the individual components scores are added together to obtain the total score
(Gezie et al., 2012).
Holistic rubrics are generally used when minor errors in the final product can be tolerated
as long as the overall focus and quality of the product remains high. These types of rubrics are
quick to make and allow for quick grading as you are only grading on the overall quality of the
product, not individual parts (Gezie et al., 2012). Generally, holistic rubrics are used when the
nature of the assessment is summative (Gezie et al., 2012). This scoring method can be used on
extended response essays where student responses are long and there are multiple responses that
could be given (McMillan, 2011).
Analytic rubrics take ample amounts of time to create as they are very detailed. Analytic
rubrics are used when a particular response is expected in order to assess the knowledge and
skills of the student (Gezie et al., 2012). This type of rubric is beneficial when teachers give a
project and are expecting particular information to be presented in the end. Because this type of
rubric is broken into components of the project it is very detailed in that each category is broken
down into scores and under each score there is a short description of what must be done to
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receive each of the particular scores. Therefore due to this detail there is a high degree of
feedback presented in this type of rubric (Gezie et al., 2012). Although it does take ample time as
the teacher must sit down and decide what a student must do in terms of organization, for an
example, to earn a score of a 0, 1, 2, or 3. In creating this type of rubric it is helpful to keep in
mind that it is better to include few categories where each level is easily differentiable than to
have so many categories you cannot differentiate the levels presented (Gezie et al., 2012).
Both analytic and holistic rubrics have their place in the classroom when it comes to
performance based assessments. However it is up to the teacher to decide what is important and
what is being measured in order to know which type of rubric is the best fit for each performance
based assessment used in the classroom.
Performance Based Task
The following performance based assessment will be incorporated into the 7th grade unit
on ratios and proportional reasoning. This performance based assessment will take the concepts
and skills taught throughout the unit to the next level as it applies these concepts and skills to real
life. Previously I have created a summative assessment in the form of a traditional paper and
pencil exam for the concepts covered in this unit. Unfortunately that type of assessment only tells
the teacher if the students know the correct answer or not, it does not do a good job explaining to
the teacher how the student arrived at a particular answer like performance based assessments
do. Therefore for this performance based assessment students will be asked to work with their
learning partners as they create an island treasure hunt map (Appendix B) using skills and
concepts they previously learned throughout the unit regarding map scale.
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Targeted Learning Objectives
This particular performance based assessment applies to the following points on the
unit’s test blueprint (Appendix A) under the content area of proportions:
Understand: Students explain how they create proportions based upon relationships
presented.
Analyze: Students will describe the proportional relationships displayed in tables, graphs,
& verbal descriptions.
Evaluate: Students will justify how they used proportions to solve problems
Create: Students will create their own map scale project using proportions to find model
and actual distances.
Targeted Common Core State Standards
The following Common Core State Standards are addressed through this particular
performance based assessment:
7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths,
areas and other quantities measured in like or different units
7. RP.2 Recognize and represent proportional relationships between quantities.
7.G.1 Solve problems involving scale drawings of geometric figures, including
computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a
different scale.
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Island Treasure Hunt Map Overview
The students will work with their learning partners to complete this performance based
assessment. I chose to have them work in these pairs because then they are automatically paired
up with someone with similar mathematical ability. This is beneficial as my class is very diverse
and I want students to show all of what they know about map scale and proportions and how they
relate to real life.
Each set of partners will work on creating an island treasure hunt map which will allow
for student creativity. Each student will be given a packet outlining the guidelines of the project
(Appendix B) as well as a grading rubric the first day we work on the project in class. Students
will be given adequate time to complete this project during class time, however they will be
given suggested guidelines as to what should be finished each day so that they will be on track to
finish on time. The first day we will read through the guidelines for part one as a class. This will
allow for students to ask questions about items they may not fully understand and it will also
allow them to see what is expected of them prior to getting started. One of the first things each
group will have to decide on is what scale to use for their map. They will be told that the scale
must use centimeters to kilometers and that they may not choose a 1 cm to 1 km or 1 cm to 2 km
scale. This scale will have to be realistic and allow for them to include all necessary pieces given
the specified guidelines.
Each map must be at least thirty kilometers wide and twenty-five kilometers long at its
longest parts. How large the island is and the shape it takes on is left up to the creativity of the
student. There must be at least three different types of terrain that are a minimum of five
kilometers apart as well as four landmarks that are also at least five kilometers apart. Their island
7th Grade Ratios & Proportional Reasoning
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must also include a water source of some type that is at least eight kilometers long. Because they
are making an island treasure map they must include an actual treasure that is exactly ten
kilometers away from any point of their water source. Completed maps must include a key on
the map depicting what the different symbols mean in regards to terrain, landmarks, and treasure
as well as include the scale of the map.
As students work through creating their maps they will be asked to complete a chart
showing all measured and actual distances from various points on their map. The actual distances
between any two points will be determined based upon the measured distance on the map and the
scale that is associated with their particular map. Students will be asked to show three of the
calculations they made in determining these distances to show they know how to set up and solve
a proportion using a scale based upon measured distances.
The last part of the project will ask students to answer several questions pertaining to
their map. A few of these questions will serve as reflection questions as they ask students to
explain and describe how they determined some of the different parts of their map and what
some of the things they did mean.
Students will be given an analytic rubric (Appendix C) that grades them on the following
components: use of scale factor, using proportions, using cross products or equal ratios,
drawing/model is done to scale, conceptual understanding, use of vocabulary, and
presentation/creativity. For each category they can receive a score between one and four. A one
would represent they showed multiple misunderstandings in the concept or failed to present it at
all. A score of four would represent that they had a strong understanding of the concept with very
minimal errors. The total score they may receive on this project is twenty-eight points, therefore
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their final grade will be the sum of each individual components awarded points divided by
twenty-eight.
Reflection
I feel that this is a good performance based assessment for this particular unit on
proportional reasoning as this represents a real world application to the unit on proportions.
Many of my previous assessments for this unit did not focus on the content area of proportions
presented in my test blueprint. Instead they focused on using proportions in real world examples
associated with applying percents and ratios. Although these concepts use proportions,
proportions are used with other content, not by themselves as they are in portions of map scale.
This performance based assessment is appropriate for my unit as it covers many of the
standards I will be covering throughout the unit. Throughout the unit I will be covering four
Common Core State Standards and this performance based assessment will cover parts of three
of them. Standard 7.RP.1 wants students to compute unit rates associated with ratios of lengths.
Students show their understanding of unit rates when they determine the scale for their map.
They then use this scale, which is a ratio in itself, to determine the lengths of distances on their
map. One way students could compute their unit rates would be to draw an island on their poster
paper and then develop a scale that would be appropriate based upon the requirements of the
project.
Standard 7.RP.2 asks students to recognize and represent proportional relationships
between quantities. Students are asked to show and discuss the proportional relationships they
see in their map both in the constructed response questions in their packet as well as when they
solve to find the measures of each of the distances between points.
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Students meet standard 7.G.1 during most of the construction of the map and while
completing the chart on page two of the guidelines packet. This standard asks students to
compute actual lengths as they solve problems that involve scale drawings. Since this project is a
scale drawing and they are computing these distances and lengths they are addressing this
standards.
This assignment will help students learn the content presented about proportions by
allowing them to apply this knowledge to a hands on task. Creating this type of visual will assist
the visual learners in deepening their understanding of the concepts presented. Many students
may have a difficult time understanding what it means to have a scale and how one would
actually go about computing a scale and using it to find useful information. It is my hope that by
allowing students to be creative in their creation they truly deepen their understanding about the
importance of proportions and why they must learn about them for reasons other than the state
said we had to.
I feel that this performance based assessment is also appropriate because it taps into the
students thinking processes as they are asked to describe, justify, and explain why they did
different things for different parts of the assignment. An analytic rubric is beneficial for this
project because there are many different parts that I as the teacher want to assess. Additionally,
there is not a broad range of responses I am looking for as many of their answers on the extended
response questions should focus around the same ideas.
Overall, this is a good performance based assessment for my seventh grade students as
they learn about ratios and proportional reasoning as it allows them to use their creativity while
applying the skills and knowledge they have learned about ratios and map scale to real world
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concepts as they create a fun map. This assessment also shows me, the teacher, if my students
truly understand the meaning of proportions and their use or if they only know how to find one,
not truly understanding them.
References
Gezie, A., Khaja, K., Chang, V., Adamek, M. E., & Johnsen, M. (2012). Rubrics as a Tool for
Learning and Assessment: What Do Baccalaureate Students Think?. Journal Of Teaching
In Social Work, 32(4), 421-437.
Hancock, D. R. (2007). Effects of Performance Assessment on the Achievement and Motivation
of Graduate Students. Active Learning In Higher Education, 8(3), 219-231.
McMillan, H., James. (2011). Classroom Assessment: Principles and Practice for Effective
Standards-Based Instruction. Boston, MA: Allyn & Bacon.
Parke, C. S., & Lane, S. (1996). Learning from performance assessments in math. Educational
Leadership, 54(4), 26-29
7th Grade Ratios & Proportional Reasoning
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Appendix A
Ratios and Proportions Test Blueprint
Proportional
Reasoning
Unit
Ratios
Proportions
Applying
Percent
Totals
Remember
Understand
Apply
Analyze
Students
will define
what it
means to
have a
ratio.
Students will
explain the
purpose of a
rate.
Students will
write “real”
world ratios.
3/ 3%
2/ 2%
Given a
table, graph,
verbal
description,
or chart
students will
identify the
rate shown.
5/ 5%
8/ 8%
Students will
describe the
proportional
relationships
displayed in
tables,
graphs, &
verbal
descriptions.
Evaluate
Create
Students
will
evaluate
ratio/rate
problems.
Students will
create a
ratio/rate
problem of
their own to
solve.
10/ 10%
5/ 5%
Students
will justify
how they
used
proportions
to solve
problems.
Students will
create their
own map
scale project
using
proportions
to find
model and
actual
distances.
Students
will
describe
the
similarities
between
creating a
ratio and
creating a
proportion.
Students
explain how
they create
proportions
based upon
relationships
presented.
Students will
write “real”
world
problems
that require
proportions
to be solved.
5/ 5%
3/ 3%
5/ 5%
10/ 10%
4/ 4%
12/ 12%
Students
will recall
how to
convert
between
fractions,
decimals,
and
percents.
Students will
explain their
thought
processes
when finding
percentages
with or
without a
proportion.
Students will
demonstrate
how they
think about
using
percents in
real life in
relation to
tax, tip,
discounts,
interest, etc.
Students will
use given
information
to solve for
percentages
and other
missing
information.
Students will
create their
own
situations to
determine
items such
as tax, tip,
discounts,
interest,
commission,
etc.
3/ 3%
5/ 5%
3/ 3%
10/ 10%
Students
will assess
when it is
appropriate
to use a
proportion
and when
they should
use the
percent
equation to
solve
problems.
11/ 11%
10/ 10%
13/ 13%
28/ 28%
2/ 2%
16/ 16%
Totals
33 /
33%
39/
39%
28/
28%
5/ 5%
32/ 32%
100/
100%
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Appendix B
Island Treasure Hunt
Due: ___________________________
Goal: Create an island using a scale of ___ cm = ___ km
**You may NOT use a scale of 1cm:1km or 1cm:2km**
Part 1:
o Name your island.
o Include a key with landmarks, terrains, water sources, the treasure, and a scale.
o The island must be at least 30 km wide and 25 km long.
o Include three types of terrain (mountains, jungle, quicksand, etc.) that are at least
five kilometers apart.
o Include four landmarks (graveyard, a tree, a monument, etc.) that are at least five
o kilometers apart.
o Include a water source that is at least eight kilometers long.
o Include an “X” where the treasure is located exactly ten kilometers away from a
water source.
o Name and label all landmarks, terrains, water sources, and treasures.
Part 2:
•
Fill out the table to calculate the distances between various landmarks on your map.
Round to the nearest tenth.
•
Answer the questions following the table. Be sure to include vocabulary terms!
•
Keep your work neat and organized.
•
Review the rubric for how your island will be graded.
•
Be CREATIVE and HAVE FUN!!!
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Fill in the table below. Show both measured and actual distances in each box. Use the next
page to show your work on how you found the selected distances.
Distances
Landmark
1
Landmark
2
Landmark
3
Landmark
4
Terrain
1
Landmark
1
Landmark
2
Landmark
3
Landmark
4
Terrain 1
Terrain 2
Terrain 3
Water
Source
Treasure
What scale did you use?__________________________
What was your scale factor? ______________________
Terrain
2
Terrain
3
Water
Source
Treasure
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Display your work for the following three table distances. Be sure to explain where the
numbers came from for each scenario.
Landmark 1 to Landmark 2:
Landmark 2 to Landmark 3:
Landmark 3 to Landmark 4:
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Question Sheet
1. Explain how you created the scale you used for your island based upon the requirements?
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
2. Explain your strategy of setting up and solving the proportions on the previous page.
Make sure you include the words scale, cross products, units, and actual distance in your
response.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
3. Describe the proportional relationship presented in your map. Why is this beneficial to
someone reading your map?
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
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4. On your map draw a path that will lead you to the treasure. This path should start from
the landmark furthest away from the treasure. The path you take must stop at every
landmark on your map before reaching the treasure. On the lines below, write out how you
would tell a friend to travel from the first landmark to the treasure using directions and
actual distances. For example: “If you start at the monument you must walk 4.6 miles
northeast to reach the apple tree.”
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
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Appendix C
Name: ____________________________ Rubric for Island Treasure Hunt Scale Drawing Project Total___/28
4
Use of Scale
Factor
Scale factor is used
correctly for all
calculations.
Using Proportions
Correctly set up
proportions for scale
drawings.
Using Cross
Products or Equal
Ratios
Demonstrates the
ability to use cross
products or equal
ratios accurately to
solve proportions.
Measurements are
correctly rounded to
the nearest tenth
with 0 to 1 calculation
errors.
An appropriate scale
was used for the
drawing. All
landmarks, terrains,
water sources, and
treasure were
included. Model is
drawn accurately with
0 to 1 minor errors.
All questions were
answered fully and
completely showing a
strong understanding
of the concepts.
Strong details.
Answers to questions
included appropriate
math vocabulary.
Drawing/Model is
Done to Scale
Conceptual
Understanding
Use of Vocabulary
Drawing
Presentation and
Creativity
Finished product is
creative, neat,
organized, colored,
and includes all
required information.
3
2
1
Scale factor is used
correctly for some
calculations. 1-2
mistakes.
Proportions are set
up correctly for some
calculations. 1-2
mistakes.
Uses cross products
or equal ratios to
solve proportions.
Shows all work and
rounds accurately to
the nearest tenth. 1
to 2 calculation
errors.
Scale factor is used
correctly for some
calculations. 3-5
mistakes
Set up proportions
that are incorrect
for scaled drawings.
Scale factor is used
correctly for some
calculations. 5+
mistakes
Fails to use proportions
to show change of
scale.
Uses cross products
or equal ratios to
solve proportions.
Shows all work and
attempts to round.
3 to 5 calculation
errors.
Fails to use cross
products or equal ratios
to solve proportions.
Does not attempt to
round answers. Shows
no work or has missing
work. More than 5
calculation errors.
An appropriate scale
was used for the
drawing. Shows at
least 8 of the
required landmarks.
Model is drawn
accurately with 2-3
errors.
An appropriate
scale was used for
the drawing. Shows
at least 7 of the
required landmarks.
Model is drawn
accurately with 4-5
errors.
Fails to use an
appropriate scale for
the drawing. Shows
fewer than 6 of the
required landmarks.
Model has more than 5
errors.
All questions were
answered showing a
good understanding
of the concepts. Has
1 mis-understanding.
Some details.
Answers to questions
included some math
vocabulary with minor
errors. Using wrong
term or definition
All questions were
answered showing
some understanding
of the concepts but
no details were
included.
Answers to
questions included
minimal math
vocabulary or
contains major
errors.
Lacks organization,
missing 2-4 pieces
of information. Not
colored.
Fails to show any
understanding of the
concepts. More than 3
misunderstandings.
Finished product is
creative, neat,
colored, organized,
and is missing 0-2
required pieces of
information.
Fails to include any
vocabulary in written
responses.
Fails to show any
organization, is not
colored, and is missing
more than 4 pieces of
information.