Meeting the Challenge:
Ohio Graduation Test for Mathematics
A Sampler of Items for Ohio’s Teachers
Copyright © 2002, Ohio Department of Education
Table of Contents
Introduction
3
Sample Items with Annotations
6
Number, Number Sense and Operation Standard
6
Measurement Standard
8
Geometry and Spatial Sense Standard
10
Patterns, Functions and Algebra Standard
12
Data Analysis and Probability Standard
14
Mathematical Processes Standard
16
Tips for Teachers
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
18
2
Introduction
This document is designed as a tool for teachers and students. It contains information about the Ohio
Graduation Test (OGT) for mathematics. Sample test questions and answers illustrate the various item
types that will be included in the assessment and are representative of the items on the actual test. Tips
for teachers and strategies for preparing for the OGT are also provided.
What mathematics content and skills are assessed by the OGT?
The OGT mathematics test assesses content knowledge and skills in using and applying mathematics.
The assessment focuses on solving problems that require the use of mathematics that is appropriate
for the end of tenth grade. Most items and tasks have real-world contexts, including the use of practical
applications, real data, and numbers often associated with situations and problems encountered in the
workplace and in daily life.
The Ohio Graduation Test is aligned with Ohio’s Academic Content Standards for K–12 Mathematics
and specifically with the benchmarks for grades 8–10. All items and tasks are directly linked to and
assess those benchmarks. A listing of those benchmarks is found on pages 160 through 163 of the
Academic Content Standards for K–12 Mathematics and also can be accessed on the Ohio Department
of Education web site at http://www.ode.state.oh.us/Academic_Content_Standards/acsmath.asp.
Every item on the Ohio Graduation Test for mathematics assesses a benchmark for one of the five
standards related to major topics within mathematics, which are:
ƒ
Number, Number Sense and Operations (approximately 15% of the test)
ƒ
Measurement (approximately 15% of the test)
ƒ
Geometry and Spatial Sense (approximately 20% of the test)
ƒ
Patterns, Functions and Algebra (approximately 25% of the test)
ƒ
Data Analysis and Probability (approximately 25% of the test)
Assessment of the Mathematical Processes benchmarks are embedded within the items and tasks
developed for the five content standards.
What types of items are used on the Ohio Graduation Test?
Three types of items are used on the Ohio Graduation Test: multiple-choice, short-answer and
extended-response.
Multiple-choice items require students to select the correct response from a list of four options. The
mathematics test includes 36 multiple-choice items.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
3
Short-answer and extended-response items require students to generate a written response. A shortanswer item requires a brief response, usually a few sentences or a numeric solution to a
straightforward problem. An extended-response item requires students to solve a more complex
problem or task and to provide a more in-depth response. Students are typically asked to show their
work or calculations, explain their reasoning and justify the procedures used.
Short-answer items may take up to five minutes to complete, and student responses receive a score of
0, 1 or 2 points. Extended-response items may require 5 to 15 minutes to complete, and responses
receive a score of 0, 1, 2, 3, or 4 points.
Each form of the mathematics test includes 36 multiple-choice items, 5 short-answer items (one for
each content standard/reporting category) and 2 extended-response items. Every content standard is
assessed by at least one extended-response item over the course of three operations forms of the
mathematics test. The test will include 43 items with a total value of 54 points.
A more detailed description of the distribution of item types and points can be found on the ODE web
site at http://www.ode.state.oh.us.
Will levels of understanding continue to be used on the Ohio Graduation Test for
Mathematics?
Careful attention will continue to be given to insure mathematics items and tasks assess the level of
mathematical thinking or cognitive demand that students may use when responding to items and tasks.
However, items will no longer be classified by levels of understanding; i.e., conceptual understanding,
knowledge and skills, and application and problem solving.
Each mathematics item and task for the Ohio Graduation Test is classified based upon its level of
complexity. Each level of complexity describes the nature of the expectations of an item or task – low
complexity, moderate complexity or high complexity.
The levels of complexity categories used by Ohio for item and test form development are aligned with
those in the Mathematics Framework for 2005 for the National Assessment of Educational Progress
(NAEP). The focus is on what the item asks of the student, and each level includes aspects of
reasoning, performing procedures, understanding concepts, or solving problems. The levels are
ordered so that items at a low level represent performance of simple procedures, understanding of
elementary or basic concepts, or solve simple problems. Items classified as high complexity require
students to reason about more sophisticated concepts, perform complex procedures or solve nonroutine problems.
Brief descriptions of the levels of complexity used in item development include:
Low Complexity: Items rely heavily on recall and recognition of facts, definitions and procedures.
Items typically specify what the students are to do and often involve carrying out a specified, routine
procedure. (Approximately 25% of the items)
Moderate Complexity: Items require more interpretation of a problem or situation and choice
among alternative solution strategies than low complexity items. Students are expected to make
decisions about what to do, using informal reasoning and problem-solving strategies. The solution
process ordinarily requires more than one step. (Approximately 50% of the items)
High Complexity: Items require more sophisticated analysis, planning, and reasoning in more
complex or non-routine problem situations. Students are often asked to think in an abstract or
sophisticated way and to justify their reasoning and solution process. (Approximately 25% of the
items)
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
4
These categories are used for item and test construction purposes only. Student performance data is
not reported by levels of complexity. The order of the levels of complexity is not intended to imply that
mathematics is learned or should be taught in a similar or prescribed order. The use of levels of
complexity for item and test construction insures items and test forms assess an appropriate balance in
both content and the range or variety of ways for understanding and using mathematics.
It should be noted that level of complexity is not directly related to item format. Additional information
and examples can be accessed by selecting “Mathematics Framework for 2005” on the publications
page on the web site for the National Assessment Governing Board at http://www.nagb.org.
What reference materials and tools can students use while taking the mathematics test?
A mathematics reference sheet is included in the test materials. Examples of information provided on
the reference sheet are common area and volume formulas, the quadratic formula, and definitions for
basic trigonometric ratios.
Scientific calculators are provided for student use while taking the mathematics test. Only the stateprovided calculator can be used during testing. Students are not allowed to use any other calculator.
Support materials are provided to familiarize students with the operation and functions of the calculator.
Provision of a specific calculator for use on the OGT is not intended to limit the types of calculators
used within mathematics classrooms. Decisions requiring acquisition and use of more advanced
technologies (e.g., graphing calculators and CBLs) at or below grade ten should not be based upon the
calculator policy for this test.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
5
Sample Items with Annotations
The sample items illustrate how some of the grades 8–10 benchmarks for each of the five content
standards and for the mathematical processes standards will be assessed on the Ohio Graduation
Test. Multiple-choice, short-answer and constructed-response items are included in the samples.
A discussion of each item includes the correct answer and highlights the knowledge and skill it is
intended to assess. The level of complexity is also identified for each item. Items are organized by
standard.
These items come from a number of sources, including Ohio’s item development process and released
items from national and state assessments. The items are representative of the kinds of items that may
appear on the Ohio Graduation Test.
It is important to remember that the sample items that follow represent a small portion of the
knowledge and skills measured by the Ohio Graduation Test. The full range of content and
expectation of Ohio’s benchmarks for grades 8–10 and items assessing those benchmarks is
not reflected in this document. Additional resources and actual items from OGT test forms will be
released in future years.
Number, Number Sense and Operations Standard
Sample multiple-choice item:
25.
Which point, R, S, T, or U on this number line
is closest to – 7 ?
A.
R
B.
S
C.
T
D.
U
Source: Ohio HSGQE Practice Test, 2000
Explanation:
Students must identify the relative position of – 7 , or –2.6, on a number line. The square root can be
approximated as between 2 and 3, which eliminates points R and U (options A and D). Point T is closer
to –2 than –3. Students have access to a calculator which can be used to approximate the value of –
7 or can use informal strategies to determine that – 7 is closer to
–3 than –2. The correct answer is point S (choice B).
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
6
This question has been classified as Moderate Complexity. The item assesses Benchmark D: Connect
physical, verbal and symbolic representations of integers, rational numbers and irrational numbers
(students must connect the symbolic representation for – 7 to its relative position on the number line);
and Benchmark H: Find the square root of perfect squares, and approximate the square root of nonperfect squares (students may use informal strategies or the calculator to approximate the – 7 .)
Sample constructed-response item: Short-answer item
Tracy said, “I can multiply 6 by another number and get an answer smaller than 6.”
Pat said, “No, you can’t. Multiplying 6 by another number always makes the answer 6 or larger.”
Who is correct? Give a reason for your answer.
Give mathematical evidence to justify your answer.
Source: Released Item, 1992 National Assessment of Education Progress
Explanation:
Tracy is correct as multiplying 6 by a number less than 1 will result in an answer that is less than 6. The
focus of this item is number and operation sense, including the effect of the size of numbers and
operations performed on using those numbers. Students can justify their answer in a variety of ways
including using counter-examples.
This item assesses Benchmark F: Explain the effects of operations on the magnitude of numbers. The
level of complexity for this item is Moderate as students must interpret a simple argument, use informal
reasoning and provide a justification for their answer.
Items on the Ohio Graduation Test are formatted according to Ohio’s Style Guide. Students are always
directed to respond in the Answer Document followed by task posed in a declarative manner. For
example, this question might be posed as “In your Answer Document, determine who is correct and
show work or provide an explanation to support your answer.” Items from sources outside Ohio are
presented as they appeared on the test or materials from which they have been drawn and may not
conform to the Ohio Style Guide used for formatting the mathematics items developed for the Ohio
Graduation Test.
A Special Note about Assessment of Mathematical Processes: The short-answer item above
includes embedded assessment of Mathematical Processes Benchmark D: Apply reasoning processes
and skills to construct logical verifications or counter-examples to test conjectures and to justify and
defend algorithms and solutions. The linkages to specific Mathematical Processes benchmarks have
been identified for this item and the two items in the Mathematical Processes section beginning on
page 15. These three items are illustrative of how mathematical processes are embedded within the
assessment of the benchmarks for the five content-related standards.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
7
Most constructed-response items and some multiple-choice items developed for the Ohio Graduation
Test will include embedded assessment of mathematical processes. For example, items and solution
processes may require a mathematical model or representation (Benchmarks A, C and E), a problemsolving strategy (Benchmarks A and H), reasoning skills (Benchmark D), and communication skills
(Benchmarks F, G and H).
Measurement Standard
Sample multiple-choice item:
15.
A graphic designer is working on a new soup label.
The rectangular label will completely cover the lateral surface of the can
using as little paper as possible. What are the width and length (to the
nearest centimeter) of the label?
A.
11 cm × 7 cm
B.
11 cm × 18 cm
C.
11 cm × 22 cm
D.
11 cm × 38 cm
Source: Ohio HSGQE Practice Test, 2000.
Explanation:
The answer is 11 cm x 22 cm (option C). The item requires students to analyze the problem context
and illustration provided. Students should recognize that the dimensions of a label using as little paper
as possible would be approximately the height and the circumference of the soup can. The key
dimension needed to select a correct response is the circumference as all answer options include the
height of the can (11 cm) as the width of the label. The circumference of the can can be found using the
formula provided on the Reference Sheet or can be approximated as three times its diameter or at least
21 cm.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
8
The item represents Moderate Complexity as it requires interpretation of a visual representation and
using information from the figure to solve the problem. This item assesses Benchmark E: Estimate and
compute various attributes, including length, angle measure, surface area and volume, to a specified
level of precision.
Sample constructed-response item:
31.
In the accompanying diagram, x represents the length of a ladder that is
leaning against the wall of a building, and y represents the distance from
the foot of the ladder to the base of the wall. The ladder makes a 60° angle
with the ground and reaches a point on the wall 17 feet above the ground.
Find the number of feet in x and y.
Source: Released Item, New York Regents A Exam, August 2002.
Explanation:
Students may use a variety of approaches to find the length of the ladder (x) and the distance between
the ladder and the base of the building (y). The distance between the ladder and building, 9.814954576
or ~9.8, can be found using the information on the Reference Sheet, such as applying the definition of
tangent, tan 60° =
17
, or by using the information given and the ratio of the lengths of the legs for a
y
30°-60°-90° triangle, 17 = y 3 . The length of the ladder, 19.62990915 or ~19.6, can also be found by
applying trigonometric ratios, such as the sine or cosine, or by using the relationships between the
lengths of the sides of a 30°-60°-90° triangle. The Pythagorean Theorem can also be used to find the
length of the third side of the triangle formed by the ladder, ground and building once the length of one
of the missing sides has been found. The scoring of constructed-response items gives full “credit” for a
number of alternate “paths” or solution processes.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
9
This item assesses Benchmark D: Use proportional reasoning and apply indirect measurement
techniques, including right triangle trigonometry and properties of similar triangles, to solve problems
involving measurements and rates. The level of complexity for this item is Moderate Complexity as
information must be retrieved from a figure and used to solve a problem requiring multiple steps.
Items on the Ohio Graduation Test are formatted according to Ohio’s Style Guide. This question would
appear in a different format on Ohio’s test. The illustration would be located immediately following the
first sentence in the item stem. The second sentence would appear under the illustration. The question
might be posed as “In your answer document, find the distance between the ladder and the building, y,
and the length of the ladder, x. Show work or provide an explanation to support your answers.”
Geometry and Spatial Sense Standard
Sample multiple-choice item:
33.
Ryan and Kathy each drew a triangle with an angle of 20
degrees. Under which condition would the triangles be
similar?
A.
if both are right triangles
B.
if both are obtuse triangles
C.
if the triangles have the same area
D.
if the triangles have the same perimeter
Source: HSGQE Practice Test, 2000
Explanation:
Understanding the concept of similarity and the characteristics of triangles is the focus of the item. The
correct answer is A as similar triangles must have the same angle measures. One measure, 20
degrees, is given, and option A establishes that both triangles have a right or 90 degree angle and, by
extension, a 70 degree angle. Option B is incorrect as both triangles could be obtuse triangles, but not
similar; e.g., a 20°-60°-100° triangle and a 20°-120°-40° triangle. Both C and D can be eliminated as
triangles having the same area or perimeter are not similar.
This question has been classified as Low Complexity as it requires recall or recognition of a fact,
property or term. The item assesses Benchmark B: Describe and apply the properties of similar and
congruent figures; and justify conjectures involving similarity and congruence. The focus of this item is
on the first part of this benchmark, describe and apply properties of similar and congruent figures.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
10
Sample constructed-response item: Short-answer item
Use the diagram below to answer question 16.
16.
a.
On the grid provided in your Student Answer Booklet, copy the diagram shown above.
Then transform the shaded “L” in the first quadrant by using the following sequence of
steps:
Step I.
Step II.
Step III.
Reflect the “L” over the x-axis.
Rotate the result of Step I clockwise 180° about the origin.
Translate the result of Step II three units up to its final position.
As you transform the shaded “L,” draw and label the image for each of the three steps.
b.
Describe a transformation with fewer than three steps that would achieve the same result
as the three steps in part a.
Source: Released Item, The Massachusetts Comprehensive Assessment System:
Release of Spring 1999 Test Items
Explanation:
A response receiving maximum points shows a thorough understanding of transformations by
performing the reflection, rotation and translation accurately. The new coordinates of the vertices of the
image after the sequence of steps are (–3,4), (–5,4), (–5,5), (–4,5), (–4,7) and
(–3,7). (Coordinates are provided to identify the location of the final image in this document; however,
students are not required to provide coordinates, only draw and label the images in the answer
document.) The result of each step in part a of the task is drawn accurately and clearly labeled. The
series of less than three transformations described in the response to part b transforms the shaded “L”
to the same position as the three steps in part a; e.g., translate the “L” up three units and reflect the
result over the y-axis. Partial points are earned for progress towards completing the steps and parts of
the task.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
11
The level of complexity for part a is Low Complexity as each step involves performing a specified
procedure. The task in part b requires representing a situation mathematically in more than one way,
which is a characteristic of items classified as Moderate Complexity. The overall complexity for the item
would be Moderate Complexity. The item assesses Benchmark F: Represent and model
transformations in a coordinate plane and describe the results.
Patterns, Functions and Algebra Standard
Sample multiple-choice item:
17. A portion of the graph of the function
y = 2x2 – 7
is shown on the grid below.
9
8
7
6
5
4
3
2
1
0
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
-4
-5
`
-6
-7
-8
-9
For which other value of x does y equal 1?
A.
–1
B.
–2
C.
–3
D.
–4
Source Released Item, Texas End-of-Course Exam for Algebra I, 2002.
Explanation:
The item assesses familiarity with the graphs of quadratic equations and solving quadratic equations.
The correct answer is –2 (Option B). The solution can be found by either applying a characteristic of the
2
graph of a quadratic equation (e.g., symmetry) or by solving the equation 1 = 2x – 7.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
12
The item assesses Benchmark G: Solve quadratic equations with real roots by graphing, formula and
factoring. Some solutions strategies may draw upon aspects of Benchmark E: Analyze and compare
functions and their graphs using attributes, such as rates of change, intercepts and zeros. The item
involves some interpretation and informal reasoning before making a choice among alternative
approaches or solutions and is classified as Moderate Complexity.
Sample constructed-response item: Extended-response item
44.
Members of a high school band are planning a
cruise on Lake Erie. They have two choices that
they like equally well.
For question 44, respond completely in your
Answer Document.
In your Answer Document, write and label an
equation that represents the total cost for a Song
Bird cruise and an equation that represents the
total cost for an Erie Queen cruise. Use c to
represent the total cost of the cruise for p, any
number of people.
The band wishes to choose the less expensive of
the two charter cruises. Describe a method that
can be used to make this decision. Support your
answer by showing your work or providing an
explanation.
Source: Ohio HSGQE Practice Test, 2000.
Explanation:
A solution earning maximum points includes a correct equation for each cruise; for example, c = 14p +
350 (or equivalent) for Song Bird Charter Cruise and c = 16p (or equivalent) for Erie Queen Cruise. The
response describes a method for deciding which option is less expensive with clear and accurate
supporting work. Students may use any of a number of mathematical models or ways of using the
system of equations to find the less expensive charter cruise; e.g., use a table, use a graph or solve the
system symbolically. Interpretation of the solution in the context of the situation is an important
component of the task.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
13
This item has been classified as High Complexity as planning, analysis and mathematical justification
for the solution are required. The task assesses Benchmark H: Solve systems of linear equations
involving two variables graphically and symbolically.
Data Analysis and Probability Standard
Sample multiple-choice item:
37.
A computer password consists of four
characters. The characters can be one of the
26 letters of the alphabet. Each character may
be used more than once. How many different
passwords are possible?
A.
104
B.
14,950
C.
358,800
D.
456,976
Source: Ohio HSGQE Practice Test, 2000.
Explanation:
The focus of this item is finding the number of possible four letter passwords. The correct response is
456,976 as there are 26 possible letters for each of the four positions or 26 x 26 x 26 x 26 possible
passwords. A number of strategies can be used to find the solution, such as informal reasoning and
counting techniques.
The task assesses Benchmark H: Use counting techniques, such as permutations and combinations, to
determine the total number of options and possible outcomes. The item is classified as Low Complexity
as the correct answer can be found by performing a specified procedure. This classification is further
supported by the availability of a calculator during testing.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
14
Sample extended-response item:
49.
A movie producer is considering two different endings for a movie. To decide which ending is
better, the producer randomly selected two groups of people to watch each ending. There were
200 people in each group. The two groups rated the movie endings on a scale of 1 to 100. The
box-and-whisker plots below show a summary of their results.
Complete the following in the Answer Book:
•
Which ending had a higher median rating? Use mathematics to justify your answer.
•
Which ending had a wider range of ratings? Use mathematics to justify your answer.
•
Based on the data shown in the box-and-whisker plots, the movie producer decided that
the first ending was better than the second ending. Is this a valid conclusion to make from
the data given? Use mathematics to justify your answer.
Source: Released Item from Maryland High School Assessment Program,
2001 Algebra/Data Assessment, Public Release, Fall 2001
Explanation:
The tasks included in this extended-response reflect multiple concepts and skills in data analysis. The
first two components ask students to compare two sets of data using specific characteristics or
components based upon box-and-whisker plots. The second ending has the higher or greater median
(a median of 78 compared to a median of 72). The first ending has the wider or greater range (a range
of 53 to 96 or 43 compared to a range of 52 to 90 or 38). There are a variety of ways students can use
mathematics to justify their answers to these questions.
The data given in the box-and-whisker plot does not support the movie producers conclusion as the
median (rating for which half of those surveyed rated the movie lower and half rated the movie higher)
is 78 for the second ending and only 72 for the first ending. The “box” indicates that 25% of those
surveyed after viewing the first ending gave the ending a rating of less than 60 and 75% of those
surveyed gave the ending a rating of 73 or less. These values are lower than the corresponding
quartiles for the second ending. Students can use a variety of methods for using mathematics to show
or explain why the data does not support the producer’s conclusion, such as organizing the information
in a table or providing a narrative, written explanation.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
15
The first two components, comparing the median and range for the endings, involve recalling the
definition or characteristics of box-and-whisker plots, which represents a low level of complexity. The
complexity of those tasks is increased when a mathematical justification is provided. Making a decision
about the producer’s conclusion involves retrieving and interpreting information from the visual displays.
The item may be classified as Moderate Complexity as the justification is based upon information
readily available in the box-and-whisker plot.
The item assesses aspects of multiple benchmarks, including Benchmark A: Create, interpret and use
graphical displays and statistical measures to describe data; e.g., box-and-whisker plots, histograms,
scatterplots, measures of center and variability; Benchmark D: Find, use and interpret measures of
center and spread, such as mean and quartiles, and use those measures to compare and draw
conclusions about sets of data; and Benchmark E: Evaluate the validity of claims and predictions that
are based on data by examining the appropriateness of the data collection and analysis.
Mathematical Processes Standard
Sample multiple-choice item:
24.
The diagram shows a carpenter’s square that is used to
measure riser height and tread length. A carpenter has been
asked to replace a staircase with one that is less steep. The
carpenter could
A.
increase the riser height leaving the tread length the
same.
B.
increase the tread length leaving the riser height the
same.
C.
increase the tread length and the riser height
proportionally.
D.
decrease the tread length and the riser height
proportionally.
Source: Released Item, The Massachusetts Comprehensive Assessment System,
Release of May 1998 Test Items
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
16
Explanation:
The correct answer is B as increasing the tread while leaving the riser height the same will reduce the
slope (ratio of rise over run) of the staircase. An increase in the length of the tread while keeping the
riser constant will reduce the angle of inclination (the angle formed by a line segment connecting the
outer edge of the tread to the top of the riser, which is the same as the angle formed by the a line
connecting the outer edge of each tread from the bottom to the top of the staircase.)
The item is classified as Moderate Complexity. Selecting a correct answer requires interpreting a visual
representation and comparing statements using informal reasoning. The item assesses Measurement
Benchmark D: Use proportional reasoning and apply indirect measurement techniques, including right
triangle trigonometry and similar triangles, to solve problems involving measurements and rates; and
Mathematical Processes Benchmark B: Apply mathematical knowledge and skills routinely in other
content areas and practice situations.
Sample Constructed-response item: Short-answer Item
7.
A gold bracelet was sale priced at 50% off the
regular price at a certain department store. Then,
for the weekend only, the sale price was reduced
an additional 15%. A customer wanted to
purchase a bracelet, but became confused when
he did not receive a 65% discount off the regular
price.
For question 7, respond completely in
your Answer Document.
In your Answer Document, explain why the
discount should NOT have been 65%.
Source: Ohio HSGQE Practice Test, 2000
Explanation:
A response earning maximum points clearly and accurately explains why the additional discount of 15%
applied to the sale price of 50% off the regular price is not the same as a 65% discount on the regular
price. There are several approaches and supporting work and/or explanations that are valid. For
example, one is to calculate the percent of discounts (57.5%) resulting from the compound or multiple
discounts. Another approach is to use an example, such as finding the discount price of a $100
bracelet, to show why the discount should not be 65%. A narrative explanation or diagram can also be
used to represent the conditions in the problem situation and to compare the discounts.
The item assesses Mathematical Processes Benchmark H: Locate and interpret mathematical
information accurately, and communicate ideas and processes and solutions in a complete and easily
understood manner and Number, Number Sense and Operations Benchmark G: Estimate, compute
and solve problems involving real numbers, including ratio, proportion and percent, and explain
solutions. The item has been classified as High Complexity. The response requires abstract reasoning,
planning and creative thought. Student must analyze the assumptions made in the problem context and
explain or justify a solution.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
17
Tips for Teachers
What can teachers do to help students?
Strategies for teachers in preparing students for success on the Ohio Graduation Test include:
•
Become familiar with the benchmarks for grades 8–10 and the indicators that represent
progress towards achieving those benchmarks. All teachers, across the K–12 program, need
to be cognizant of the specific benchmarks and the level of expectation for the new test.
Information about the test also needs to be shared with students and parents.
•
Discuss interpretations of benchmarks. A critical component of preparing students for the test
is having a balanced mathematics program in which students become proficient with basic skills,
develop conceptual understanding, and become adept at problem-solving. Engaging in discussion
of the skills, concepts, and applications of the mathematics content to be assessed is important
when designing an instructional program.
•
Identify required learning experiences based on benchmarks. The mathematics course of
study and implementation should be reviewed and adjusted, as needed, to insure ALL students
have access to the full curriculum, or core preparation, regardless of past achievement or course
options taken. The depth and breadth of the content and instructional experiences are critical in
determining the role a specific course or grade level makes in addressing the standards. Course
“titles” alone are not sufficient for determining if students have access to the curriculum and
instruction needed to demonstrate the benchmarks and for success on the Ohio Graduation Test.
•
Match instructional strategies and materials to identified student learning experiences.
Instructional strategies and materials across the program and within grade levels/ranges should
be reviewed for alignment and consistency in emphasis and implementation.
•
Develop a “critical eye” for selecting or developing assessment strategies and items.
Multiple assessment strategies for assessing student progress and achievement should be
included within the mathematics program. Particular attention should be given to incorporating
student generated-response items and tasks into classroom assessment. Short-answer and
extended-response items should be encountered regularly in instruction and assessments at the
classroom and district levels. Professional development in classroom assessment should include
issues related to identifying and writing good assessment items and tasks, implementing new
assessment strategies, and scoring student responses and making inferences about student
learning. Assistance for implementing new assessment techniques can be found through print and
Internet resources. A variety of sample and released items international, national and state
assessments are available via the internet.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
18
Some specific tips and suggestions for teachers include:
Ö
Provide lots of practice in reading and analyzing word problems.
•
•
•
Ö
•
Include student-generated response items in every assignment and assessment.
Promote self-assessment by providing a guide or checklist for use by students as they
construct responses.
Involve students in creating and using task-specific rubrics to evaluate sample responses to
classroom tasks.
Develop comfort and skill using a calculator as a tool.
•
•
•
•
Ö
Select word problems that develop and practice process skills such as communication,
problem-solving, mathematical reasoning, and representation.
Promote high-quality writing and responses to short-answer and extended-response
items.
•
•
Ö
Focus student attention on reading and thinking about a word problem as a whole, or
complete “package,” including diagrams and illustrations, for meaning and context rather than
depending on word hints or “keywords” to prompt them as to what operation(s) will be needed
to solve the problem.
Practice solving word problems, particularly multiple-step problems, should be an integral part
of day-to-day classroom instruction and assessment.
Develop skill in deciding when it is more efficient to compute mentally, when an estimate is
needed, or when a calculator result is reasonable.
Practice number sense skills such as predicting the effects of operations prior to performing
calculations and making decisions about the reasonableness of results.
Estimate solutions mentally to problems based on real-life situations.
Provide access to and permit use of calculators in instruction and assessment whenever
computation is not the objective of the lesson.
Follow a yearlong instructional plan.
•
•
•
Balance mathematics content and processes by including opportunities for students to
develop communication, problem-solving, mathematical reasoning, and representation skills
while learning new content.
Embed effective practice of basic skills within and through the learning of new concepts and
skills.
Closely monitor “pacing” of instruction so important concepts are not “left over” at the end of
the school year.
To Obtain Further Information on
The Ohio Graduation Tests, contact the Office of Assessment 614-466-0223
Academic Content Standards, contact the Office of Curriculum and Instruction 614-466-1317
Ohio Department of Education
Offices of Curriculum, Instruction and Assessment
25 S. Front Street, Mail Stop 507
Columbus, OH 43215-4183
The Ohio Department of Education does not discriminate on a basis of race, color, national origin, sex, religion,
age, or disability in employment or the provision of services.
Ohio Graduation Test for Mathematics—Test Sampler, November 2002
19