Download AIM Resource Manual

Articulation & Integration of Mathematics
(AIM)
Resource
Prepared for the
NYC Office of Adult and
Continuing Education
by TERC, Inc.
with contributions from NYCOACE Instructional Facilitators
and Teachers
Fall 2011
AIM – Articulation and Integration of Mathematics
AIM Development Team
Patrick Cravillion, Evette Corbin, Steve Meyerson, Linda Pelc, Mary Jane Schmitt
(TERC), John Belcher (TERC)
AIM Algebra Workshop Planning and Co-Facilitation Team
Patrick Cravillion, Mara Komoska, Steve Meyerson, Linda Pelc, Mary Jane Schmitt
(TERC), John Belcher (TERC)
AIM Geometry and Measurement Workshop Planning and Co-Facilitation Team
Patrick Cravillion, Steve Meyerson, Linda Pelc, Mary Jane Schmitt (TERC), John
Belcher (TERC)
AIM Data, Statistics and Probability Workshop Planning and Co-Facilitation Team
Patrick Cravillion, Rebecca Guillen-Samuels, Diana Raissis, Mary Jane Schmitt
(TERC), John Belcher (TERC)
AIM Number Sense and Operations Workshop Planning and Co-Facilitation Team
Patrick Cravillion, Rhonda Naidich, Mary Jane Schmitt (TERC), John Belcher (TERC)
AIM Workshop Participants
Betty Aderman, Evette Corbin, Patrick Cravillion, Palamona Ferris, Harry Goldstein,
Rebecca Guillen-Samuels, David Gutmann, Cynthia Hanratty, Sylvester Jaward,
Diann Jenkins, Gita Kaufman, Mara Komoska, Steve Meyerson, Rhonda Naidich,
Katie Naplatarski, Jolan Ostane, Linda Pelc, Diana Raissis, Beverly Segers,
Stephanie Varner-Mnere, Karen Wald
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Table of Contents
What is AIM?
4
How was the AIM resource developed?
5
The AIM Resource – Overview and Guide to Use
To Teach or Not to Teach to the Test
6
7
How the demands of the TABE and GED align with the EFF Use Math Standard
and Performance Continuum
8
The Content Strands
10
Patterns, Functions and Algebra Across All Levels
11
What does Patterns, Functions and Algebra Across All Levels mean?
11
How do Patterns, Functions and Algebra show up on the TABE?
11
How do Patterns, Functions and Algebra show up on the GED?
12
Suggested Materials and Resources for Patterns, Functions and Algebra
Geometry and Measurement Across All Levels
What does Geometry and Measurement Across All Levels mean?
13
13
How does Geometry and Measurement show up on the TABE?
How does Geometry and Measurement show up on the GED?
13
14
Suggested Materials and Resources for Geometry and Measurement
Data, Statistics and Probability Across All Levels
15
What does Data, Statistics and Probability Across All Levels mean?
15
How does Data, Statistics and Probability show up on the TABE?
15
How does Data, Statistics and Probability show up on the GED?
16
Suggested Materials and Resources for Data, Statistics, and Probability
Number Sense and Operations Across All Levels
17
What does Number Sense and Operations Across All Levels mean?
17
How does Number Sense and Operations show up on the TABE?
17
How does Number Sense and Operations show up on the GED?
18
Suggested Materials and Resources for Number Sense and Operations
AIM and the Four Big Ideas
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3
Moving Forward
20
Pulling the Pieces Together
20
Building Upon the Work Done Thus Far
21
Appendices
23
EFF Alignment Analyses Tables
TABE Analysis – Patterns, Functions and Algebra
Patterns, Functions, Relations (PFR) to Understanding and Using Symbols (UUS)
TABE Analysis – Geometry and Measurement
Geometry and Measurement – What’s on the GED?
TABE Analysis – Data Analysis, Statistics, and Probability
Data, Statistics and Probability GED Analysis
TABE, GED, and OPT Analyses – Number and Operations Sense
Suggested Materials and Resources for the Content Strands
List of Resources
The Four Big Ideas
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What is AIM?
AIM (Articulation and Integration of Mathematics) is a resource for helping
OACE teachers:
 Articulate mathematics learning expectations between the BE 1/2, BE
3/4 and GED levels.
 Integrate key OACE mathematics curricular, instructional and
assessment resources – including EMPower, Equipped for the Future
(EFF), Math Problem Solver, Key to … , the TABE and the GED.
 Develop student understanding in each of the four mathematics
strands (Number Sense and Operations; Patterns, Functions and Algebra;
Data Analysis, Statistics, and Probability; Geometry and Measurement)
across all levels.
What concepts and skills can a GED teacher expect to have been developed in
BE classes? What experiences can a BE3/4 expect were part of a BE1/2 class?
How does algebra develop through the levels? Geometry? Number sense?
Data? How do the TABE Levels E, M, D and the GED connect to one another?
Having resources that facilitate a shared understanding of the concepts and skills
with which students should have familiarity, competency, or mastery over at the
different levels is critical to making informed, strategic instructional decisions,
decisions that make sense across the arc of a student’s learning journey.
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How was the AIM resource developed?
In the spring, 2010, OACE instructional facilitator Patrick Cravillion and Schools 1
and 2 teacher, Evette Corbin met with Mary Jane Schmitt and John Belcher from
TERC to examine items from the TABE E, M and D and from GED Official
Practice Tests (OPTs) to determine the demands of these assessments for
Algebra, as it appears at all of the levels. Finding the process eye-opening, the
team discussed the value of having a larger group of teachers, representing
different levels of mathematics instruction, participate in such a process and use
this analysis work as a basis for identifying how the mathematics teaching and
learning materials and resources, in use at OACE, can be more cohesively used
to address these demands, as well as the expectations of the EFF Use Math to
Communicate Standard and Performance Continuum.
During the winter and spring of the 2010-11 school year, a group of
approximately 20 teachers and Instructional Facilitators from OACE’s first two
EMPower training cohorts met over the course of four Articulation and Integration
of Mathematics Instruction (AIM) sessions with Mary Jane Schmitt and John
Belcher from TERC to analyze the items on the TABE and GED OPT, to discuss
the implications of the findings, and to identify how various adult mathematics
instructional materials being used in OACE classrooms address the demands of
the TABE and GED. As well, members of the group had initial conversations
about how the demands of the TABE and GED align with the expectations of the
EFF Use Math to Communicate Standard and Performance Continuum.
The AIM sessions were co-planned and co-facilitated, collectively, by OACE
Instructional Facilitators Patrick Cravillion, Linda Pelc, Rhonda Naidich, Rebecca
Guillen-Samuels, and Diana Raissis and OACE GED Teacher Steve Meyerson.
The work produced in these sessions was followed up upon and organized into
this version of the AIM resource by Patrick Cravillion, Mary Jane Schmitt and
John Belcher, with input from Steve Meyerson and Linda Pelc.
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The AIM Resource – Overview and Guide to Use
This AIM resource is organized according to each of the four mathematics
strands – Number Sense and Operations; Patterns, Functions and Algebra; Data,
Statistics and Probability; Geometry and Measurement. The section for each
strand includes:
(1) Identification of key content and skills demands at the different levels, as
reflected in the TABE E (indicating BE 1/2 level demands), in the TABE M
(indicating BE 3/4 and GED demands), and in the TABE D and OPTs (indicating
GED demands).
(2) How EMPower, Math Problem Solver, Key to… and other resources OACE
teachers use address the particular content strand at the different levels.
In referring to Suggested Resources, teachers should keep in mind the importance of
ensuring a balance between procedural fluency and conceptual understanding. Adult math
education resources typically emphasize procedural practice. Consequently, in the present
versions of the Suggested Resources charts for the different strands there is an abundance
of materials emphasizing procedures in the “Other Resources OACE Teachers Currently
Use” column. Moving forward, The AIM Development Team encourages teachers to seek
out (and/or develop) resources, including those online, that equip students to approach a
wide variety of problems in a multiple ways (e.g., visually and symbolically) and that make
effective use of the Four Big Ideas.
The AIM resource is intended to be dynamic in the ways that it is used. The
information provided in this version of the resource is meant to be more guide
than prescription. The work produced thus far provides the contours of a math
“scope and sequence” for ABE and GED instruction, with the expectation that
refinements and reevaluations will occur over time.
Ways to use
 Quick reference for lessons to go to (in EMPower, Key to …, Math
Problem Solver, etc.) in order to address core areas of content at the
appropriate level
 A way of helping determine what to prioritize at the level(s) you are
teaching.
 Resource for identifying and sharing additional, supplemental
(complementary) mathematics teaching and learning resources teachers
have found to be effective.
 Resource for guiding communications with your colleagues to help
coordinate what is taught in your respective classes -- to minimize nonproductive redundancies, to fill in gaps, etc..
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 A way of getting familiarized with or deepening understanding of the TABE
and GED.
 A way of getting a better understanding of how EFF relates to EMPower,
the TABE, the GED, and other instructional and assessment resources.
 A resource to be used as a guide for newly hired teachers.
To Teach or Not to Teach to the Test
Why analyze the TABE tests and the GED? Clearly, there is value in
understanding what is covered on these tests. Knowing what your students are
being held accountable for is critical information. This information is valuable for
guiding instruction but can be problematic if used to set the parameters for what
students are exposed to. Though having a clearer understanding of what is
covered on the tests can be helpful in making decisions about what to emphasize,
there is no intent behind the analyses of the TABE and GED for encouraging
teachers to teach to the test.
That being said, one can consider what it might really mean to teach to the test.
The analyses show the students need to have facility in using different
representations (pictures, tables, symbols, graphs, words) for different content.
Students need to have strategies for determining whether or not an answer is
reasonable. Students need to understand clearly what they are being asked to do
or to find. Students need to develop mental math skills. And so on. So even if
someone decided to teach to the test, the teaching involved is more than just
going through a content checklist.
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How the demands of the TABE and GED align with the EFF
Use Math Standard and Performance Continuum
During the summer 2011, John Belcher, Mary Jane Schmitt and Patrick Cravillion
reviewed the work produced in AIM workshops across the year. One significant
piece of work that remained was to show how the demands of the TABE and
GED align with EFF. The team also felt that it would be important to determine
the most helpful “grain size” for analyzing the TABE and GED. They developed a
new set of TABE and GED analysis templates using content examples of the
content strands at the different levels, based upon the EFF Use Math to Solve
Problems and Communicate Curriculum Framework, (developed by Equipped for
the Future, along with the Oklahoma Department of Education Lifelong Learning
Section and the Dollar General Literacy Foundation) along with the GED content
descriptions.
In some cases, the team adjusted the language of the content examples of the
EFF Use Math Framework to reflect more clearly what the example might look
like on assessments like the TABE or GED. In other cases, in doing the analyses,
the team recognized that a student who was competent with a particular EFF
content example would be equipped to handle a particular test item (since the
demands of the content example subsumed the demands of the problem(s) in
question).
Generally speaking, the team found that the demands of the TABE and GED
draw upon a range of content and skills that can be effectively developed in the
purposeful, life roles’ contexts encouraged by EFF. They also found that most of
the TABE M and TABE D Applied Math items fall within the content and skill
suggested for EFF levels 3 and 4, in the EFF Use Math Framework. The team
found that GED test items, typically, are more aligned with EFF level 5 and, to
some extent, level 6 expectations.
Analyses of TABE and OPTs based on EFF Use Math Performance
Continuum and GED Content Area Descriptions
Patterns, Functions and Algebra
BE 1/2
BE 3/4 and GED
GED
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Geometry and Measurement
BE 1/2
BE 3/4 and GED
GED
Data, Statistics and Probability
BE 1/2
BE 3/4 and GED
GED
Number Sense and Operations
BE 1/2
BE 3/4 and GED
GED
A recommended area of work in the upcoming year is to examine further ways in
which alignment and dissonance exist between the EFF Use Math Framework
and TABE and GED demands. This can inform more targeted thinking about how
to cover more effectively the range of mathematics learning needs and purposes
of OACE students.
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The Content Strands
Number Sense and Operation; Patterns, Functions and Algebra; Geometry and
Measurement, and; Data, Statistics and Probability
The sections that follow are organized according to four content strands: 1)
Number Sense and Operation, 2) Patterns, Functions and Algebra, 3) Geometry
and Measurement, and 4) Data, Statistics and Probability. Though it can be
helpful to think about the mathematics being taught as falling more heavily in one
category versus another, at the same time, it is important to stay mindful of the
interrelationships amongst strands. Work in Data, Statistics and Probability
involves, naturally and fundamentally, the use of percents, decimals, and
fractions. Geometry and Measurement can entail use of formulas and
understanding growth patterns (e.g., how perimeter and area are affected by
changes in the lengths of sides). The development of algebraic thinking occurs in
the study of Number Sense and Operations, through recognizing patterns
(counting by 2’s and 5’s, for example) and making sense of procedures for
manipulating numerical expressions. Being mindful of these interrelationships
helps in recognizing opportunities for teaching across strands which can assist in
addressing the time issues that arise from aiming to cover discrete pieces of
content.
The order of the math content strands in the AIM resource (e.g., beginning with
Patterns, Functions and Algebra) isn’t meant to imply an order of emphasis or
importance. These decisions should be guided by what you know of and/or learn
about your students’ needs. However, the AIM Development Team did feel that
by presenting Patterns, Functions and Algebra as the first content strand in this
resource, it does reinforce the idea that opportunities exist to teach all strands at
all levels.
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Patterns, Functions and Algebra Across All Levels
What does Patterns, Functions and Algebra Across All Levels mean?
In initial AIM planning sessions, the planning team decided to begin with
considering algebra demands for students. They felt that it was important to
challenge the strict sequential way of thinking about mathematics learning, in
which algebra is considered to be a subject that students take only after
mastering the fundamentals of number sense and operations. In this line of
thinking, unless and until students “know their multiplication tables” and are able
to add, subtract, multiply and divide fractions, they will not be prepared to “do
algebra.” The team sought to identify algebra demands as they appear as early
as BE 1/2, so that teachers and students would recognize the need and
opportunities to develop algebraic thinking over time.
How do Patterns, Functions and Algebra show up on the TABE?
Initial Process
The AIM Development Team used the American Council on Education (ACE)
Content Description for Algebra, Functions and Patterns on the GED
(http://www.acenet.edu/Content/NavigationMenu/ged/etp/math_test_descriptio.ht
m) to develop a template for analyzing the TABE for algebra demands at the
different levels. The team determined that the content description clustered into
two areas: Patterns, functions, relations; Understanding and Using Symbols. The
team used this template to analyze items on the TABE 10 E, M and D tests.
At the AIM Algebra workshop in December, participants used the template to
analyze the TABE themselves to see if they came up with similar results as the
AIM planning team. (TABE – Patterns, Functions and Algebra)
Findings of TABE Analyses based upon GED Algebra Content Descriptions
Among the key findings:
 There were a significant number of algebra items at all of the test levels, E,
M and D.
 On the Mathematics Computation tests on the TABE E, M, D, there were
almost no algebra items, based on the GED algebra content descriptions.
(There was one algebra item on the TABE 10D Computation test.)
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 Most of the TABE algebra items fell into the “Understanding and Using the
Symbols” category (e.g., simplifying expressions or evaluating formulas)
which emphasized procedures and symbol manipulation skills.
 The algebra items on the TABE reflect “more than the study of the last 3
letters of the alphabet” (i.e., the problems involved more than solving for x,
y, or z).
How do Patterns, Functions and Algebra show up on the GED?
Process
The AIM Development Team used the same template used for analyzing the
TABE to identify and categorize algebra items on the Official Practice Tests
(OPTs) for the GED. Again, they determined how items fell into the categories of
Patterns, Functions, Relations and Understanding and Using Symbols. At the
AIM Algebra workshop in December, participants examined different OPTs and
to see how their choices matched up with those of the planning team.
Participants also discussed how they would place selected items along a
continuum -- Patterns, Functions, Relations (PFR) to Understanding and Using
Symbols (UUS).
Findings
Among the key findings:
 The GED classifies ratio and proportion problems as algebra, even very
easy ones.
 When workshop participants examined selected items according to how
they lay across a continuum – from Patterns, Functions and Relations
(emphasizing conceptual understanding) to Understanding and Using
Symbols (emphasizing procedures) -- they determined that even for
procedure-based test items, students are better equipped if they are able
to draw from a conceptual understanding base (for example, to know
which possible answers make sense or not).
Suggested Materials and Resources for Patterns, Function and
Algebra
BE 1/2
BE 3/4
GED
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Geometry and Measurement Across All Levels
What does Geometry and Measurement Across All Levels mean?
It is helpful to think of geometry, not, primarily, as a course in a sequence of
courses. Typically, viewed in this fashion, geometry is available to more
advanced students who have mastered arithmetic and successfully navigated
their way through algebra (Algebra 1). However, since we constantly deal with
space, shapes, and measurement in our daily lives, geometry is a branch of
mathematics with places of access for many students who might think of
themselves as “non-mathematical.” As well, just as students employ geometric
thinking in many aspects their lives, the skills and understandings developed and
used in geometry are engaged in all of the other mathematics strands at all levels.
For example, visual models are helpful in thinking about what one-half looks like.
Mentally rotating and/or “flipping” (reflecting) a shape to determine whether or not
it is the same as another shape anticipates the more formal developments of the
concepts of similarity and congruence later on. There are many opportunities, at
all levels, for students to develop and use the skills and understandings of
geometry.
How do Geometry and Measurement show up on the TABE?
Process
The AIM-Geometry and Measurement workshop was the last of the AIM
workshops held during the 2010-11 school year. The planning team considered
the advantages and disadvantages of having a more or less itemized template,
including concerns that too many item categories would feel like a content
checklist and that too few categories, though beneficial for reinforcing content
connections, might be less helpful in shedding light on what to emphasize in
instruction. The template they developed seemed to have the right “grain size” for
where the AIM group was in its process. The planning team used the template to
analyze Geometry and Measurement items on the TABE 10 E, M and D Applied
Mathematics tests. At the AIM-Geometry and Measurement workshop, the
planning team gave participants slips of paper with item numbers of the TABE
problems they identified as geometry and measurement problems. Participants
were asked to place the problems in the appropriate categories on the analysis
template. Discussing how particular items might fit into more than one category
and into another strand helped the group think more deeply about how geometry
and measurement skills and understanding appear on the TABE.
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Findings of TABE Analyses based upon GED Geometry and Measurement
Content Descriptions (and other sources)
Among the key findings:
 Members of the AIM group were surprised at the number of geometry
items, given that geometry instruction is often given short shrift in the lower
instructional levels.
 The geometry tested is primarily 2-D geometry.
 A significant number of items involve interpreting scales, meters and
gauges.
 Geometry and measurement skills and understanding are critical to
answering problems from the other strands. For example, reading and
interpreting number lines is related to reading and interpreting scales and
meters. Many of the data problems involve circle graphs. Recognizing
benchmark fractions representations in circle graphs is important in solving
these problems.
 Many items require visualization skills – i.e., rotating and translating
shapes, recognizing similar and congruent shapes, and recognizing
benchmark fractions.
 Problems involving time were common (in particular, having to calculate
differences between times)
How do Geometry and Measurement show up on the GED?
Participants used the analysis template to analyze various OPTs for geometry
and measurement items. They were able to compare their choices with those of
the planning team for PA, PD, and PG.
Among the key findings:
 A greater variety of geometry and measurement items are found on the
TABE in comparison to the items on the OPTs.
 Problems that involve finding, using, and interpreting the slope of a line,
the y-intercept of a line, and the intersection of two lines may be classified
as geometry (instead of algebra) problems by the GED Testing Service.
Suggested Materials and Resources for Geometry and Measurement
BE 1/2
BE 3/4
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GED
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Data, Statistics and Probability Across All Levels
What does Data, Statistics and Probability Across All Levels mean?
There are many opportunities to develop Data, Statistics and Probability skills
and understandings across all levels. At the lower levels, in addition to engaging
data at a basic level (e.g., extracting information from a chart), students can
begin to practice more sophisticated skills, such as looking at displays of data
and coming up with comparative statements. Learning to recognize benchmark
fractions, decimals and percents in a bar or circle representation helps build a
foundation for interpreting data displayed in bar and circle graphs. Understanding
of statistics such as range, mean, mode, median develops over time, and is done,
most effectively, by including experiences that provide a feel for these ideas (e.g.,
EMPower “Line Up By Size” activity) to support formal rules and procedures for
finding them. Investigating real life issues and questions through collecting,
organizing, representing, and interpreting data and analyzing the data using the
appropriate statistics can be done across the levels. At the higher levels,
students can be expected to make more complex and sophisticated arguments
using data and statistics.
How do Data, Statistics and Probability show up on the TABE?
Process
The AIM planning team used the American Council on Education (ACE) Content
Description for Data, Statistics and Probability on the GED
(www.acenet.edu/Content/NavigationMenu/ged/etp/math_test_descriptio.htm) to
develop a template for analyzing the TABE for data, statistics and probability
demands at the different levels. The team discussed the how best to categorize
items and settled upon the categories of Data (e.g., organizing data), Statistics
(e.g., finding the mean), and Probability (e.g., permutations). The team used this
template to analyze items on the TABE 10 E, M and D Applied Mathematics tests.
At the AIM – Data, Statistics and Probability workshop in January, participants
used the template to analyze the TABE themselves to see if they came up with
similar results as the AIM planning team.
Findings of TABE Analyses
Among the key findings:
 There are very few problems that involve statistical measures (mean, etc.)
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 Most of the data items involve reading or extracting data. No real
interpretation involved.
 Only a few probability problems. Problems that involve making statements
about an outcome being likely, unlikely, etc. are typical. Also, typical are
simple probability problems (either discrete situations, like dice, or
continuous situations, like a spinner).
 Items with circle graphs at all levels of the TABE.
 Items that involve reading schedules and calendars appear on the TABE E.
How do Data, Statistics and Probability show up on the GED?
Data, Statistics and Probability analyses of OPTs
Findings
Among the key findings:
 One of the OPTs (PA) emphasizes problems involving statistical measures
(mean, median, line of best fit). The problems involving mean ask the
student to think of what numbers are needed to arrive at the desired mean
(instead of simply computing the mean).
 Sometimes the student is asked to match a “story” to a graph.
 Typically, the probability problems (only a few across the OPTs) involve
combinations.
 GED uses problems that require reading schedules to extract information.
Suggested Materials and Resources for Data, Statistics and
Probability
BE 1/2
BE 3/4
GED
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Number Sense and Operations Across All Levels
What does Number Sense and Operations Across All Levels mean?
Number Sense and Operations is the mathematics strand that involves
understanding properties of numbers, ways of representing them, ways of
relating them and ways of manipulating them. Typically, more opportunities are
given for students to build number “sense”, i.e., getting a bone-deep feel for
numbers, than are given to build “operation sense.” What do addition,
multiplication, subtraction and division (and exponentiation) mean? When does it
make sense to use which operation? What do these various operations look like
in our day-to-day lives?
At all of the levels, students need practice with representing numbers and
operations in various ways (pictures, diagrams, using manipulatives) and using
them in different contexts (working in discrete and continuous settings, in
measurement, money, in patterns). Difficulty can be adjusted by using “friendly
numbers” at the lower levels and more challenging numbers (e.g., negative
numbers, non-benchmark fractions) at higher levels. Number sense and
operations skills and understanding can be practiced and developed in the other
strands using the same strategy of choosing friendlier or more challenging
numbers.
How do Number Sense and Operations show up on the TABE?
Process
The planning team for the AIM-Number workshop in January 2011 decided that it
would be helpful to have breakdowns for the types of numbers and operations
that typically appear on the TABE Computation and Applied Tests and on the
GED (based on the OPTs). They created charts to capture information such as:
 How many problems involve fractions? What are the fractions?
 How many problems involve decimals? What is the nature of the number?
(money?, tenths?, hundredths?...)
At the workshop, participants worked in teams, based upon the levels they teach,
to examine the TABE E, M, and D Computation and Applied Tests.
Findings of TABE Analyses
For the TABE E:
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 On the TABE 10E Computation Test, there are no problems involving
fractions or percents. All of the problems involving decimals used money;
these problems require only addition and subtraction.
 On the TABE 10E Applied Test, 43 out of 50 problems involve visual
literacy (i.e., reading charts, tables, diagrams, pictures, etc.). Of the 10
problems involving decimals, only one is in a non-money context. There
are no percents problems.
For the TABE M:
 On the Computation Test, are no percents problems. There are 8 problems
with fractions – no multiplication or division. The non-money decimal
problems use tenths and hundredths and involve addition, subtraction, or
multiplication (no division). There are no signed numbers.
 On the Applied Test, most of the fractions are benchmark fractions. There
are no percents problems. All of the decimal problems involve money.
Most of the problems require visual literacy.
For the TABE D:
 On the Computation Test, the fraction problems involve addition,
subtraction, multiplication and division. The test includes percent problems
(involving multiplication) and problems involving addition, subtraction,
multiplication or division of decimals. There are problems involving the
addition, subtraction, multiplication or division of signed numbers.
 On the Applied Test, there is a nearly equal distribution of problems
involving whole numbers, fractions, decimals, and percents. Most of the
decimal problems are non-money. Many of the problems on the Applied
Test call upon visual literacy.
How do Number Sense and Operations show up on the GED?
Among the key findings, from analyzing OPTs:
 Typically, about 10% of the problems involve fractions.
 There are a greater number of problems involving decimals and percents
(compared to fractions).
 Typically, the students need to figure out which operation or combination of
operations to use in solving a problem.
Suggested Materials and Resources for Number Sense and
Operations
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BE 1/2
BE 3/4 (Whole Numbers)
BE 3/4 (Rational Numbers)
GED
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AIM and the Four Big Ideas
OACE mathematics professional development in support of implementing
EMPower has been organized around the “four big ideas” of:
 Communication
 Connections
 A Richer Definition of Mathematical Proficiency
 All Strands at All Levels
These big ideas have served as lenses for developing the AIM resource. The
analyses of the TABE and GED demonstrate the need to teach all mathematics
content strands at all levels for students to be effectively equipped to engage
these assessments. The “Suggested Resources and Materials” sections inform
how EMPower, complemented with other resources, can be used strategically to
develop student skills and understandings, at the different levels, across
mathematics content areas.
In addition, the TABE and GED analyses suggest the efficacy of using a variety
of communication strategies, including student-to-student communication,
making connections, within formal mathematics as well as to familiar,
meaningful life situations, and using a richer definition of mathematical
proficiency, beyond learning procedures, in order for students to be prepared for
the variety of problems, calling for a variety of problem solving strategies, they
will encounter. Indeed, anticipated changes to the GED, the new version of which
will be aligned to the Common Core Standards, will call even more upon students
to show robust understanding beyond manipulating symbols and using
procedures.
The Four Big Ideas can be valuable in developing or modifying lessons and/or
reflecting upon a lesson as actually experienced in the classroom.
The EFF Math Standard and Performance Continuum fully aligns with the Four
Big Ideas.
OACE AIM Resource, Fall 2011
23
Moving Forward
Pulling the Pieces Together
Again, though the AIM workshops were organized and the analyses done
according to individual content strands, an overarching goal was to identify
opportunities to connect ideas, content and skills across strands. Below are
some selected examples of key content and skills, from different content
strands, emphasized on the TABE and the GED. They are chosen to reiterate
the importance of students being given opportunities to investigate all of the
content strands at all of the levels.
BE 1/2
 Emphasize visual representations of benchmark fractions. Students should
be able to represent benchmark fractions (halves and quarters) in various
ways (pizza model, bars, tiles, and gauges and scales [see below]). They
will be able to draw upon these skills in interpreting different
representations of data (e.g., circle graphs), placing fractions on a number
line, and understanding angle measurement (e.g., a quarter turn is a
rotation through 90 degrees).
 Have students practice reading scales, meters, gauges, etc. These
measurement skills provide/reinforce a foundation for reading number lines
(and plotting points).
 Use real life contexts to practice time measurement and reading (and
making) schedules.
 Structure opportunities for students to develop an intuitive feel for
perimeter and area, including recognizing where and how these ideas
show up in their lives and understanding why perimeter is measured in
linear units and area is measured in square units. Having a secure feeling
for perimeter and area is instrumental in making sense of procedures for
finding these quantities computationally. Later, students will be able to
draw upon their understanding/feel for the distinction between perimeter
and area in reasoning about linear vs. nonlinear growth patterns.
 Have students practice recognizing and creating simple patterns (using
numbers and shapes).
BE 3/4
 Emphasize lessons in which students generate and organize data. This will
help in matching “stories” to graphs. Providing opportunities to represent
data in bar graphs and circle graphs and developing flexibility in going
OACE AIM Resource, Fall 2011
24




back and forth between these representations is also critical. Asking
students to make comparative statements about data using benchmark
fractions and percents is also valuable practice.
Provide practice with decimals in a variety of contexts – not just money.
Time measurement (who won the race?) is an effective way of
practicing/understanding place value. Investigating the perimeter of
different shapes using sides with decimal measurements provides a
context for adding and subtracting (given a perimeter find the side)
decimals. Investigating the area of rectangles using dimensions in
decimals provides a context for multiplying and dividing decimals.
Help students recognize and understand equivalencies between
benchmark fractions, decimals and percents. Have students look for
examples of how these turn up in their daily lives (e.g., “half off” and “50%
off”). Students should make charts of these equivalencies; the charts can
be used as references/tools in developing facility in moving amongst
fractions, decimals, and percents.
Provide opportunities for students to locate signed numbers, including
benchmark fractions and decimals, on the number line.
Provide proportional reasoning practice in different situations, such as
“good” vs. “bad” reproductions, recipes, and conversions (of measurement
units, money, etc.).
GED
 Emphasize decimals and percents (less work with fractions) – ordering
decimals, converting between decimals and percents, finding percents of
arbitrary quantities, multiplying and dividing decimals.
 Have students investigate measures of central tendency (mean, median
and mode) in order to develop further their intuitions about the meanings of
these measures and about what they tell us about data sets. Various
strategies for adjusting the difficulty level of problems involving mean and
median include using decimal numbers and/or solving open-ended mean
and median problems (e.g., given a set of test scores, what score could a
student get on the next test to have an average score of at least _____?).
 Have students investigate signed numbers in different contexts (e.g.,
temperature scales, balances and debts, direction) in order to make sense
of the rules for adding, subtracting, multiplying and dividing signed
numbers. Students will find it helpful to have a variety of signed number
models from which to draw.
 Have students practice the use of variables in different contexts – for
example, on the GED, there are problems in which students need to be
able to determine the area of a rectangle with side lengths expressed in
variables (e.g., length, x units; width, 2x units).
OACE AIM Resource, Fall 2011
25
 Provide opportunities for students to determine rules for input/output tables
and express them algebraically.
 Use familiar life experiences, such as choosing a phone plan or
determining car rental costs, to develop understandings of slope and
intercept points in graphs of linear functions.
Building Upon the Work Done Thus Far
The present form of this resource is just a start, a capturing of thoughtful work
done collaboratively by a group of OACE teachers and instructional facilitators,
with support from TERC consultants. At the AIM-Geometry workshop, the last
AIM workshop for the 2010-2011, participants shared ideas for moving forward.
Among the suggestions for moving forward:
 Introduce more teachers to the process of analyzing the TABE (or the
CASAS) and OPTs. This is a powerful professional development activity
and an effective way of teachers understanding what is on these tests and
of the learning for which students are being held accountable.
 Conduct AIM workshops/work sessions in which teachers and instructional
facilitators work together to develop and/or modify lessons and activities,
using the EFF Use Math… Standard and the “Four Big Ideas”. This
includes adapting EMPower lessons to different levels and modifying
lessons from conventional adult education mathematics resources that are
more procedures based.
 Continue to add to the lists of Suggested Resources for the content
strands at the different levels. Emphasize materials that involve effective
use of the Big Ideas.
OACE AIM Resource, Fall 2011
26
Appendices
OACE AIM Resource, Fall 2011
27
Patterns, Functions and Algebra at BE 1/2
TABE E Analyses Based on EFF Use Math Performance Continuum
and GED Content Area Descriptions
TABE
TABE
9E
10E
Read and interpret symbolic information
Recognize, explain and/or use the meaning of +, - , x ,
÷ , = symbols
30, 39
Solve simple number sentences and explain
relationships within “families” of numbers, such as 3 +
__ = 5, 5 – 3 = __ , 5 = __ + 3
23, 43
Create and use algebraic expressions and equations to
model situations and solve problems
14
Interpret and apply very simple patterns, functions,
and relationships
Recognize and apply simple number patterns such as,
counting by 2s, 5s, or 10s
21, 29
Identify, describe in simple terms, and use basic
properties of operations (0 as the additive identity, 1 as
the multiplicative identity, commutative property)
Use visual information (pictures, diagrams, etc.) to
solve problems
OACE AIM Resource, Fall 2011
28
Patterns, Functions and Algebra at BE 3/4 and GED
TABE M and TABE D Analyses Based on EFF Use Math Performance Continuum and GED
Content Area Descriptions
TABE
TABE
9M
10M
18
18, 41
TABE 9D
TABE
10D
Read and interpret symbolic information
Show repeated multiplication for simple whole numbers using
exponents (such as 32 = 9)
Recognize, explain and/or use the meaning of +, - , x , ÷ , =
symbols
8, 36
Apply order of operations to solve equations and to evaluate
expressions
Solve simple number sentences and explain relationships
within “families” of numbers, such as 3 + __ = 5, 5 – 3 = __ , 5
= __ + 3
Evaluate expressions
42
7
44
48
16, 40,
26, 42
47
Write statements of equality or inequality (such as 3 > 4 – 3)
OACE AIM Resource, Fall 2011
22
29
Solve a variety of equations and inequalities
TABE
TABE
9M
10M
TABE 9D
20
10D
33
8
Create and use algebraic expressions and equations to model
situations and solve problems
TABE
13, 32,
12, 40
42
Interpret and apply a variety of common patterns,
functions, and relationships
Recognize, describe, and/or generalize rules for simple,
repeating patterns
Recognize and develop repeating patterns and generalize the
relationship with a table, rule, graph, or formula
2, 46
46
21, 34
45, 46
2
45, 46
Identify, simply describe, and use common properties of
operation (associative and distributive property)
20
Generalize, in words, the relationship between quantities in a
table of amounts (including in-out tables)
Develop formulas and/or create simple linear graphs from
tables
OACE AIM Resource, Fall 2011
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30
OACE AIM Resource, Fall 2011
31
Patterns, Functions and Algebra at GED
OPTs Analyses Based on EFF Use Math Performance Continuum and GED Content Area
Descriptions
PA
PB
PC
PD
PE
PF
PG
Read and interpret symbolic information
Apply order of operations to solve equations and
evaluate expressions
Evaluate expressions and solve a variety of
equations and inequalities, including quadratic
equations and exponential functions
Solve two linear equations in two variables and
display in graphic form
3, 8, 9
9, 18,
21, 24
13
19
Use polynomials and expressions with rational
exponents
OACE AIM Resource, Fall 2011
32
Interpret and apply a wide variety of
complex patterns, functions, and
relationships
Recognize, describe, and extend patterns and
develop formulas
10
Represent patterns and relationships (including reallife situations) with algebraic expressions, formulas,
tables, or graphs
11, 19,
11, 19,
12, 16,
24, 25
20, 21
17
Describe general shape and qualities of linear
graphs, including the unit rate of change as slope of
the line and the intercept points
5, 6
11, 12,
25
Describe general shape and qualities of simple nonlinear graphs
15
21
Create and analyze a wide variety of functions and
relations, including linear and exponential functions
Analyze situations involving cost (such as
profit/loss margin) or earnings/deductions
OACE AIM Resource, Fall 2011
24
21
33
Geometry and Measurement at BE 1/2
TABE E Analysis Based on EFF Use Math Performance Continuum and
GED Content Area Descriptions
TABE
TABE
9E
10E
Read, write, interpret, and apply very simple types of
information related to measurement and geometry
Read and write time (hours and minutes, day and
months, etc.)
Read and interpret common measuring tools (rulers,
measuring cups, gauges, scales, meters, etc.) to
nearest whole unit
Identify and select appropriate units of metric and
customary measures
8
3, 49
46
Develop benchmark estimates for common measures
(such as inch, foot, yard, pound)
Identify shapes and solids
Use spatial visualization skills (e.g., rotating and
translating shapes; recognizing benchmark fractions)
10, 20
1, 6, 19,
44, 48
Recall and use a few simple mathematical procedures
Use appropriate tools to measure high-frequency
standard whole units of measure (i.e., pounds, inches,
feet, gallons, cups) to the nearest whole unit
2
Find the perimeter and area of rectangles and other 2D shapes using simple strategies
OACE AIM Resource, Fall 2011
34
Find lengths
TABE
TABE
9E
10E
24, 25
Apply simple coordinate graphing elements (positioning
of points on a scale or number line and rectangular
coordinates)
3
Show equivalent amounts of money using different
coins and dollar bills
45
Recognize and apply simple number patterns such as,
counting by 2s, 5s, or 10s
OACE AIM Resource, Fall 2011
21, 29
35
Geometry and Measurement at BE 3/4 and GED
TABE M and TABE D Analyses Based on EFF Use Math Performance Continuum and GED
Content Area Descriptions
TABE
TABE
9M
10M
37
37, 38,
TABE 9D
TABE
10D
Read and interpret symbolic information
Recognize and use commonly used standard units of
measurement to the nearest half and quarter
39, 42
Identify and select appropriate units of metric and customary
measures
33, 35
Recognize and describe polygons, including basic angle
descriptions (such as acute, right and obtuse)
3, 36
44
23
8, 33
34, 35,
38
3
Recognize and describe parts of a circle
Recognize and describe 3-D solids
30
17
17
27
Determine the different attributes of three-dimensional cubes
and rectangular solids, including volume and surface area
OACE AIM Resource, Fall 2011
36
TABE
TABE
9M
10M
24, 25
50
TABE 9D
TABE
10D
Use benchmark angles (such as 90 degrees and 45 degrees)
to estimate size of angles
Recognize and describe parallel, perpendicular and skew lines
48
Demonstrate an understanding of the coordinate graph
system, including ordered pairs
Use spatial visualization skills (e.g., rotating and translating
shapes; recognizing similar and congruent shapes;
recognizing benchmark fractions)
29
3, 9, 40,
1, 11,
2, 18, 26,
13, 30,
41
13, 22,
34
43
23, 29,
(trans.)
33
Interpret and apply a variety of common patterns,
functions, and relationships
Use appropriate tools to measure to the degree most
appropriate (e.g., nearest half or quarter unit) for the situation
37
Make simple conversions within the same measurement
system (such as inches to feet, cm to m)
34
Convert units of measure from one system to another using
informal methods (i.e., a cm is about half an inch)
OACE AIM Resource, Fall 2011
37
Use direction, distance, labels, simple scales, and symbols to
read and use maps and plans
TABE
TABE
9M
10M
26, 44
TABE 9D
TABE
10D
48
22, 23,
Determine whether two-dimensional shapes have similar
attributes and properties (e.g., Are they congruent? Are they
similar?)
29
Model and solve problems using the concepts of congruence
and similarity of geometric figures
Solve and estimate solutions to problems involving length,
perimeter, area of common two-dimensional shapes
45
18,
40, 41
20, 24
43
24, 32
Use uniform rates (e.g., miles per hour, bushels per acre) in
problem situations
Add and subtract dimensional numbers (such as 5’ 2” - 3’ 7”,
such as 5:20 AM minus 3:40 AM) to solve problems involving
measurement and geometry
Show equivalent amounts of money using different coins and
dollar bills (From EFF Level 1/2)
OACE AIM Resource, Fall 2011
14, 28,
27, 38,
19
11, 29
18
44
10
38
Geometry and Measurement at GED
OPTs Analyses Based on EFF Use Math Performance Continuum and GED Content Area
Descriptions
OACE AIM Resource, Fall 2011
39
PA
PB
PC
PD
PE
PF
PG
Read, write, interpret, and apply a wide
variety of mathematical information
related to measurement and geometry
Use the language of metric unit prefixes to describe
real-life measurements
Solve and estimate solutions to problems involving
length, perimeter, area of common two-dimensional
shapes
13
11
22
22
Measure size of angles using tools such as
protractors
Solve and estimate solutions to problems involving
angle measurement
Modified GED Geometry and Measurement
language
Use coordinates to describe geometric figures or
their transformations (e.g., rotations, reflections,
etc.)
22
Select and use multi-step mathematical
procedures
OACE AIM Resource, Fall 2011
40
Use ratio and proportion to solve problems
involving scale drawings or similar figures
14, 18,
6, 23
25
Solve problems involving congruence and
similarity of geometric figures
Use proportional reasoning to convert between
measurement systems
Use uniform rates (e.g., miles per hour, bushels per
acre) in problem situations
4
Solve problems involving perpendicularity and
parallelism
Apply the knowledge of properties of triangles (e.g.,
180 degrees in a triangle) to solve problems
17
24
11
Use the Pythagorean theorem to solve problems
involving right triangles
Predict the impact of changes in linear dimensions
on the perimeter, area, and volume of a figure
Add and subtract dimensional numbers (such as 5’
2” - 3’ 7”, such as 5:20 AM minus 3:40 AM) to
solve problems involving measurement and
geometry
(EFF Level 4)
OACE AIM Resource, Fall 2011
2
11
5, 16
41
Determine the dimensions of various common twoand three dimensional shape (i.e., length when the
area is given, or the width if the height, length, and
volume are given)
8
18
Determine the slope of a line, the y-intercept, and
the intersection of two lines
OACE AIM Resource, Fall 2011
42
Data, Statistics and Probability at BE 1/2
TABE E Analysis Based on EFF Use Math Performance Continuum and
GED Content Area Descriptions
TABE
TABE
9E
10E
Read and interpret data and statistical information
6, 7, 15,
17, 27,
Extract discrete information from simple bar graphs,
from simple circle graphs, or from a list or table
28, 34,
35, 36,
37, 49
Draw conclusions from information displayed in simple
bar graphs, from simple circle graphs, or from a list or
table
Describe concrete information found in a graphic
representation
Make relative comparisons about values on a bar graph
or other graphic representation of data (such as
“greater than” or “less than” or “about twice as much” or
“between 30 and 40”)
16, 47
Collect, organize, and represent data
Use check sheets, picture graphs, and frequency
graphs to collect and organize data based on posed
questions
OACE AIM Resource, Fall 2011
43
Use Probability
Describe the probability of an event within the range of
“likely”, “possible”, “unlikely” or “impossible”
41
Find the probability of a single outcome in a coin toss
OACE AIM Resource, Fall 2011
44
Data, Statistics and Probability at BE 3/4 and GED
TABE M and TABE D Analyses Based on EFF Use Math Performance Continuum and GED
Content Area Descriptions
TABE
TABE
TABE
10M
9D
10D
14, 15, 43
27
37
41
4, 5, 6,
11, 12,
13, 16,
21, 27,
28, 30,
32, 40
3, 8, 12,
25, 26,
31
16, 19,
20, 27,
28, 31,
35, 36
3 (line
plot), 4
(lp), 6, 7,
8, 36
13, 26,
28
21, 22
22
TABE 9M
Read and interpret data and statistical information
Identify, find and/or use the shape, range, median, mean, and mode of
data
Describe the effect of range and outliers on median and mean
Extract discrete information from a list, table, bar graph, pictograph, or
line plot
Make predictions or draw conclusions from information displayed in a
list, table, bar graph, circle graph, pictograph, or line plot
7
Make statements and numerical comparisons about relative values on a
bar or circle graph (such as “one category is three times greater than
another” or “this bar extends more than halfway between 25 and 50”)
16, 40
OACE AIM Resource, Fall 2011
45
Collect, organize, and represent data
Collect and organize categorical data based on posed question and
organize in a bar graph, or line plot
33
Collect and organize data based on posed question and represent the
information in a circle graph or stem-and-leaf plot
39
18, 26
13
Convert bar graphs into circle graphs and describe how the two are
related and what each graph represents
50
Choose sample group based upon the type of data to be collected
Use Probability
Describe the probability of an event within the range of “likely”,
“possible”, “unlikely” or “impossible”
14, 15
25
Find the probability of a single outcome in a simple concrete situation
with a very limited number of possible outcomes (i.e., the role of a die)
Find the probability of a single outcome in a concrete situation and state
it as a ratio, fraction and percent
OACE AIM Resource, Fall 2011
49, 50
17, 24,
25
21
46
Data, Statistics and Probability at GED
OPTs Analyses Based on EFF Use Math Performance Continuum and GED Content Area
Descriptions
PA
PB
PC
PD
PE
PF
PG
Read and interpret data and statistical
information
Identify, find and/or use the shape, range,
median, mean, and mode of data
Modified EFF Level 4
1, 2, 13
Extract and/or use information from a list, table,
chart or graph
Modified EFF Level 3
Make predictions, draw inferences or draw
conclusions from information displayed in a
table, chart, or graph.
Modified EFF Level 3
Use an informal line of best fit to interpret data,
make predictions and draw conclusions
OACE AIM Resource, Fall 2011
15
6, 7, 14,
8, 14,
15
15, 19
1, 2
9
4, 5, 6
47
PA
Make statements to support or refute an
argument using interpretations from data
PB
PC
PD
PE
PF
PG
25
Describe the effect of changes in the data set on
the mean and median and discuss which is more
representative of the data
Collect, organize, and represent data
Create scatterplots to compare two variables and
informally estimate lines of best fit to test
hypotheses
Use line of best fit to make predictions and draw
conclusions about data
Use Probability
Recall and use basic probability concepts to find
the probability of independent events happening
Find the probability of a single outcome in a
concrete situation and state it as a ratio, fraction
and percent
EFF Level 4
OACE AIM Resource, Fall 2011
48
PA
Find combinations and permutations
OACE AIM Resource, Fall 2011
PB
PC
PD
PE
PF
PG
20
49
Number Sense and Operations at BE 1/2
TABE E Analysis Based on EFF Use Math Performance Continuum and
GED Content Area Descriptions
TABE
TABE
9E
10E
Read, write, and interpret very simple types of
mathematical information and concepts and apply to
real-life and theoretical problems
Whole Numbers/Integers
Write numbers in words (e.g., another way to write
5200 is five thousand two hundred)
9, 32
Use place value to create equivalent representations of
numbers through three digits (such as 45 is the same
as 4 tens and 5 ones or 20 + 20 + 5)
Determine the relative size of numbers of up to four
digits
Use the inverse relationship between addition and
subtraction to add and subtract whole numbers up to
four digits in a variety of problems including those
related to geometry, measurement and data
Choose appropriate strategies to check for
reasonableness of answers
Rational Numbers (Fractions, Decimals and Percents)
Recognize that 50% is the same as 1/2 is the same as
.5 and 25% is the same as 1/4 is the same as .25
OACE AIM Resource, Fall 2011
13
50
Recognize and use multiple representations (pizza
models, bars, tiles, etc.) of one half and one quarter
TABE
TABE
9E
10E
19
Determine one-half and one-quarter (and their
equivalent forms 50% and 25%) of an amount by
halving and estimate when amounts are close to these
benchmark fractions and percents
Add and subtract numbers involving benchmark
fractions
24, 25
Recognize that a ratio remains the same in simple
proportions using concrete representations (e.g, tiles,
pictures, diagrams, etc.)
Solve problems involving money
4, 5, 11,
12, 26
Identify the place value of digits in the context of money
(e.g., what is the value of the 2 in $2565.00?)
Use estimation to solve problems and assess the
reasonableness of an answer
OACE AIM Resource, Fall 2011
22
26, 31,
36, 40
51
Number Sense and Operations at BE 3/4 and GED
TABE M and TABE D Analyses Based on EFF Use Math Performance Continuum and GED
Content Area Descriptions
TABE
TABE
9M
10M
TABE 9D
TABE
10D
Read, write and interpret a variety of common
mathematical information and concepts and apply to reallife and theoretical problems.
Whole Numbers/Integers
2
Write numbers, up to six digits, in words
Identify the place value of digits in numbers up to millions
Use properties of numbers (magnitude and order, place value,
factors, multiples, etc.) to solve problems
Find factors and multiples
OACE AIM Resource, Fall 2011
22
30
1, 4, 6,
4, 30
22
6 (place
value)
1
52
15
Add, subtract, multiply and divide to solve a variety of
problems, including those related to geometry, measurement,
and data
Estimate solutions to problems involving addition, subtraction,
multiplication or division to determine reasonableness of
results
5, 16, 21,
5
28, 43
15 (div.),
15 (div.)
50
(mult.)
Recognize and apply negative integers in real contexts (such
as thermometers, winning and losing money, sea level, etc.)
1
Recognize and represent negative integers on a number line
Use the rules of order for all the operations
18
36
44
Rational Numbers (Fractions, Decimals and Percents)
31
Identify and use decimal place values
2
Create equivalent representations of numbers up to billion and
to the nearest thousandth
Extend benchmark fractions to equivalent decimals and
percents (1/10, 1/100, etc.)
OACE AIM Resource, Fall 2011
48
9
53
Extend bank of benchmark fractions, decimals, and percents
(1/8, 1/6, etc.) and understand how these relate on a number
line
8
42
35, 36
1
Add, subtract and multiply numbers involving benchmark
fractions, decimals and percents
38
9, 35, 42
19, 36,
20, 36,
45, 46
37, 48
(mult.),
49
24
Solve problems involving non-benchmark fractions
Solve problems involving money
13, 23,
9, 10, 43
5, 31, 42
30, 32
48
14, 38
Round whole and decimal numbers
46
Add and subtract decimals up to three places
Build on understanding of ratios, to include equivalent forms of
benchmark fractions (such as 2/4 = 1/2)
10
OACE AIM Resource, Fall 2011
5, 46
43
17
16
Find and interpret ratios
Understand how to express equivalent quantities in
percentages and as decimals and fractions
6, 10, 12,
49
4
17
54
Use benchmark fractions, decimals, and percents (such as 3/4
and 1/10) to estimate relative sizes
19, 39
7, 45
2, 12
19, 22
Apply proportional reasoning to simple, one-step problems
39
Represent and/or interpret fractions in multiple ways
Use benchmark fractions, decimals and percents (such as 1/8,
15%) to determine estimates of fractional parts and to check
for reasonableness
Use estimation (e.g., rounding, front-end, clustering, etc.)
9
8, 9
7, 16
Exponents
Show repeated multiplication for simple whole numbers using
exponents (such as 32 = 9)
(from EFF Patterns, Functions and Relationships, Level 3)
OACE AIM Resource, Fall 2011
55
Number Sense and Operations at GED
OPTs Analyses Based on EFF Use Math Performance Continuum and GED Content Area
Descriptions
PA
PB
PC
PD
PE
PF
PG
Read, write and interpret a wide variety of
mathematical information and concepts
and apply to real-life and theoretical
problems
Whole Numbers/Integers
Add subtract, multiply, and divide integers to solve
a variety of problems
3
Solve problems involving money
3
Use place value and/or the commutative,
associative, and distributive properties to create
equivalent representations of integers of any size
4
Rational Numbers (Fractions, Decimals and
Percents)
OACE AIM Resource, Fall 2011
56
PA
Add, subtract, multiply and divide fractions to solve
a variety of problems
PB
PC
PD
PE
PF
PG
8 (add
or sub.)
Add, subtract, multiply and divide decimals to solve
a variety of problems
4
21
(divide)
(alg.
prob.)
Solve a variety of problems involving percents (e.g.,
finding percents)
10, 12
6, 7, 23
3, 7, 8,
9, 19,
20
Use benchmark fractions, decimals and percents to
make estimates and to check for reasonableness
12
Build on understanding of ratios, to include
equivalent forms of benchmark fractions (such as
2/4 = 1/2)
EFF Level 3
8
17
Find ratios
Solve problems involving money (e.g. adding and
subtracting dollar amounts, finding interest)
7, 10,
13, 23
3, 12
12, 16,
23
OACE AIM Resource, Fall 2011
57
PA
Read, write, and compare fractions and mixed
numbers and decimals
Use proportional reasoning to solve a variety of
problems, including percent increase or decrease
PB
PC
PD
PE
PF
PG
10
17
1, 2, 6
Exponents
Determine square roots to solve problems such as
those related to geometry
Evaluate expressions with positive and negative
exponents to solve problems such as growth or
decay over time
OACE AIM Resource, Fall 2011
13
24
58
TABE Analysis – Patterns, Functions, and Algebra
Algebra Assessment: Form 10, TABE E, CTB, McGraw-Hill, 2003
Patterns, Functions, Relations
No Algebra related problems on Test 2
(Computational)
Understanding and Using the Symbols
No Algebra related problems on Test 2
(Computational)
Form 10E
p. 34, #21
p. 36, #29
#14, p. 32
#23, p. 34
#30, p. 36
#39 (set-up problem), p. 39
#43, p. 40
Algebra Assessment: TABE Form 10M
Patterns, Functions, Relations
#21, p. 32
#30, p. 35
#34, p. 37
#30 TABE classifies under Number
Understanding and Using the Symbols
p. 29, #8 (set-up)
p. 31, #16
p. 37, #36
p. 38, #40
p. 38, #41
P. 41, #47
p. 41, #48
#s 8, 36, 41 and 48, TABE classifies as
under Number
Algebra Assessment: TABE Form 10D
Patterns, Functions, Relations
#2, p. 27
$20, p. 33
OACE AIM Resource, Fall 2011
Understanding and Using the Symbols
Test 2, #40, p. 25
Test 3
#26, p. 35
#28, p. 36
#33, p. 37
#40 (set up), p. 39
#42, p. 40
#44, p. 40
59
Patterns, Functions, Relations (PFR) to Understanding and Using Symbols (UUS)
Continuum
Item #7
PFR
UUS
Item #9
PFR
UUS
Item #23
PFR
UUS
Item #24
PFR
OACE AIM Resource, Fall 2011
UUS
60
TABE Analysis – Geometry and Measurement
Recognize, describe or
analyze 2-D shapes
TABE – 9E
TABE -- 10E
TABE -- 9M
13
20
3, 36
Use Pythagorean
Theorem
TABE – 10M
TABE – 9D
TABE – 10D
3, 8, 33
34, 35, 38, 45
20
Length, perimeter, area
46
19
37, 44, 45
18, 24
15, 39, 40, 41
14, 28, 43
Solve problems using
perpendicularity,
parallelism, congruence
and similarity
Translations, reflections
and rotations
10, 22
1
3, 9, 24, 25
22, 23, 29, 50
10, 33
29
10, 22
1
3, 9
1, 22, 23, 29
34
Use spatial visualization
skills (e.g., rotating and
translating shapes;
recognizing benchmark
fractions)
10, 22, 29,
30
1, 44, 48
3, 9
1, 22, 23
2, 18, 26
OACE AIM Resource, Fall 2011
13, 30, 34
61
Recognize, describe or
analyze 3-D solids
34
10, 48
17
17
27, 29, 30
Find surface area,
volume
Find, use, and interpret
the slope, the y-intercept
of a line, and the
intersection of 2 lines.
Find or use rates
10
Use coordinates.
48
24, 32
Read and interpret
scales, meters, gauges,
and number lines
16, 20
3, 8
Recognize, understand
and convert units
4, 49
2, 46
Other
OACE AIM Resource, Fall 2011
33, 34, 35
37, 38, 39, 42
1, 2, 35, 36, 37,
43
1
44, 45
12, 30
23
26 (Directions)
62
Geometry and Measurement – What’s on the GED?
Recognize, describe or
analyze 2-D shapes
PA
PD
PG
17, 22
13
11, 23
Use Pythagorean Theorem
2
Length, perimeter, area
8, 11, 25
7, 18, 23
Solve problems using
perpendicularity,
parallelism, congruence
and similarity
Translations, reflections
and rotations
18
6
22
22
Use spatial visualization
skills (e.g., rotating and
translating shapes;
recognizing benchmark
fractions)
OACE AIM Resource, Fall 2011
63
Recognize, describe or
analyze 3-D solids
Find surface area, volume
Find, use, and interpret the
slope, the y-intercept of a
line, and the intersection of
2 lines.
Find or use rates
6
Use coordinates.
22
11, 12, 19
25
Read and interpret scales,
meters, gauges, and
number lines
Recognize, understand and
convert units
Other
OACE AIM Resource, Fall 2011
64
TABE Analysis – Data Analysis, Statistics and Probability
Form 9E
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
21, 23, 24, 31, 32, 35, 36,
37, 41, 42, 43, 44
Form 10 E
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
6, 7, 8, 15, 16, 18, 27, 34,
35, 37, 38, 42, 49
Form 9M
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
4, 5, 6, 7, 11, 12, 13, 16,
21, 27, 28, 29, 40
Form 10M
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
3, 4, 5, 11, 12, 13, 25, 26,
27, 28, 31, 32, 33
OACE AIM Resource, Fall 2011
Measures of Central
Tendency and Other
Statistical Measures
Probability
50
47, 48
Measures of Central
Tendency and Other
Statistical Measures
Probability
41, 47
Measures of Central
Tendency and Other
Statistical Measures
Probability
14,15
49, 50
Measures of Central
Tendency and Other
Statistical Measures
Probability
14, 15
65
Form 9D
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
16,18, 19, 20, 21, 22, 23,
24, 26, 27, 28, 29, 30, 31,
32, 35, 36, 37, 39, 40, 41,
42, 43
Form 10 D
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
Measures of Central
Tendency and Other
Statistical Measures
Probability
17, 25
Measures of Central
Tendency and Other
Statistical Measures
Probability
6
OACE AIM Resource, Fall 2011
66
Data Analysis, Statistics, and Probability GED Analysis
DSP Assessment: GED
 (GED Testing Service Test Specifications)

Data Analysis, Statistics, and Probability
Collecting, Organizing, and Displaying Data
(Including Trends and Change Over Time)
Construct, interpret, and draw inferences
from tables, charts, and graphs.
Make inferences and convincing arguments
that are based on data analysis.
 Evaluate arguments that are based on data
analysis, including distinguishing between
correlation and causation.
Represent data graphically in ways that
make sense and are appropriate to the
context.
OACE AIM Resource, Fall 2011
Measures of Central Tendency
Probability
and Other Statistical measures
Apply measures of central
 Make predictions that are
tendency (mean, median,
based on experimental or
mode) and analyze the effect
theoretical probabilities,
of changes in data on these
including listing possible
measures.
outcomes
Use an informal line of best fit to
predict from data.
Apply and recognize sampling
and bias in statistical claims.
Compare and contrast different
sets of data on the basis of
measures of central
tendency and dispersion.
67
Facilitators’ Findings
OPT Form
PA
PB
PC
PD
PE
PF
PG
Collecting, Organizing and
Displaying Data (Including
Trends and Change Over
Time)
15, 20, 21
1, 2, 7, 8, 9 14, 15, 19, 21
5, 6, 7, 16, 17, 18, 19
1, 2, 12, 14, 15, 19, 20, 21
1, 3, 21, 23, 24
5, 6, 14, 19, 20, 21
8, 9, 14, 15, 19, 25
Measures of Central
Tendency and Other
Statistical Measures
Probability
1, 2, 4, 5, 6, 13
1, 2, 25
2
15
5
9, 18
10, 20
GED
PA
PB
PC
PD
PE
Data Analysis, Statistics and Probability
1, 2, 4, 5, 13, 15, 20, 21
1, 2, 7, 8, 14, 15, 19, 21
5, 6, 7, 16, 17, 18, 19
1, 2, 12, 19, 20, 21, 25
1, 2, 9, 18, 21, 22, 23, 24
OACE AIM Resource, Fall 2011
68
TABE and GED OPT Analyses – Number Sense and Operations
TABE Analysis
Directions:
Work in teams and examine TABE tests ( 9 and 10 -- E, M, D).
Which TABE test? 9-___________ (E, M, or D) ; 10- ____________ (E, M, or D)
For the Computation section:
What did you notice about the operations and types of numbers? Anything else?
OACE AIM Resource, Fall 2011
69
For the Applied section:
How many problems involve whole numbers only?
How many problems involve fractions?
How many problems involve decimals (not money)?
How many problems involve decimals (money)?
How many problems involve percents?
How many problems involve visual literacy? (Charts, tables, graphs, pictures, diagrams)
What else do you notice about the TABE, concerning number operations and number
sense?
OACE AIM Resource, Fall 2011
70
GED Analysis
Directions:
Pair up and examine a GED Practice Test by answering the questions below.
How many problems
What are the fractions?
involve fractions? (Which
problems?)
What operation(s)?
____ Addition
____ Subtraction
____ Multiplication
____ Division
How many problems
What are the decimals?
involve decimals? (Which
problems?)
What operation(s)?
____ Addition
____ Subtraction
____ Multiplication
____ Division
How many problems
involve percents? (Which
problems?)
What are the percents?
What operation(s)?
____ Addition
____ Subtraction
____ Multiplication
____ Division
OACE AIM Resource, Fall 2011
71
How many problems involve
exponents, scientific notation,
roots? (Which problems?)
What are the exponents (scientific
notation, etc.)?
How many problems involve time?
(Which problems?)
What are the times? (How are they
represented?)
How many problems involve “naked
numbers”? (i.e., no situation or
context given) Which problems?
What are some examples?
Give examples of problems where mental math or estimation is sufficient
for finding the answer (no need to do it out on paper or use a calculator)?
What else do you notice about the GED, concerning number operations and
number sense?
Look at the GED Content Specifications. Which ones are emphasized? Which
ones, if any, are not addressed?
OACE AIM Resource, Fall 2011
72
Number Sense and Operations on the GED
OPTs Analyses
PA
PB
PC
PD
How many
Fractions…
4
2
0
4
Decimals
2
5
5
7
7
7
4
8
1
7
7
2
1
4
4
0
5
5
Decimals -Not money
Decimals -Money
OACE AIM Resource, Fall 2011
5
PD
PF
PG
2
PE
6
4
73
Percents
2
Exponents
Roots
Scientific
Notation
3
1
5
3
3
2
2
1
2
2
2
3
3
6
4
0
0
0
1
8
1
4
6
4
2
1
INVOLVE
TIME
NAKED
NUMBERS
JUST
WHOLE #
17
EST.
OACE AIM Resource, Fall 2011
3
3
10
2
74
Patterns, Functions, and Algebra
BE 1/2
Suggested Resources for Developing Patterns, Functions, and Algebra Proficiency at the BE1/2 Levels
To be sufficiently prepared to complete the BE1/2 Levels and begin the BE 3 Level, manage the mathematical demands of work and family, students should develop proficiency with the
following content topics related to Patterns, Functions, and Algebra.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
OACE Teacher Developed
Resources
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Everyday
Number Sense (ENS).
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Achieving TABE Success E
Pretest Part B: Applied Math,
Problems 8,9, and 26
Pre-Assessment
Patterns, Functions, and Algebra
Read and Interpret Symbolic Information
Ex. Recognize and explain the meaning of +, - ,
= symbols
TABE Fundamentals M Missing
Element, pp. 44-45
Ex. Solve simple number sentences and explain
relationships within “families” of numbers, such
as 3 + __ = 5, 5 – 3 = __, 5 = __ + 3
Achieving TABE Success M
Analyzing Relationships in Number
Sentences, p. 113
p. 3 - Number of the Day asks
Get It Together
p. 36 – Number Shapes 1 and p. 37
– Number Shapes 2
Achieving TABE Success E Writing
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
75
Patterns, Functions, and Algebra
BE 1/2
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
OACE Teacher Developed
Resources
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Everyday
Number Sense (ENS).
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Ex. Evaluate simple expressions
students to generate expressions,
given a number.
Algebraic Expressions, pp. 102-103
Evaluating Expressions, p. 104
Lesson 8: Picture This
Interpret and apply a wide variety of
common patterns, functions, and
relationships
Ex. Recognize and apply simple number
patterns such as, counting by 2s, 5s, or 10s
Mental Math Practice:
p. 26, Fast Actions with 10 or 100
p. 27, Fast Actions with 9 or 90
p. 47, Doubles
p. 48, Triples
pp. 62, 63, 76, Counting Up and
Down by 10’s and By What Did I
Count?
Achieving TABE Success E
Using Patterns in Multiplication
and Division, p. 98
Ex. Identify, describe in simple terms, and use
basic properties of operations (0 as the additive
identity, 1 as the multiplicative identity,
commutative property)
Ex. Simple In/Out Tables
TABE Fundamentals M Functions
and Patterns, pp. 42-43
Achieving TABE Success E
Recognizing Patterns Among
Number Sentences, pp. 100-101.
Algebra Skills Checkup, p. 105
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
76
Patterns, Functions, and Algebra
BE 1/2
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
OACE Teacher Developed
Resources
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Everyday
Number Sense (ENS).
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Ex. Use of visual information (pictures,
diagrams, etc.) to solve problems
Review and Test Practice
Achieving TABE Success E
Identifying Patterns, pp. 95-96
Achieving TABE Success E Algebra
Skills Checkup, pp. 105-106
Post-Assessment
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
77
Patterns, Functions, and Algebra
BE 3/4
Suggested Resources for Developing Patterns, Functions, and Algebra Proficiency at the BE3/4 Levels
To be sufficiently prepared to complete the BE3/4 Levels and begin the GED level, manage the mathematical demands of work and family, and to anticipate preparation for post-secondary
readiness, students should develop proficiency with the following content topics related to Patterns, Functions, and Algebra.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the
lessons below are from Seeking
Patterns, Building Rules:
Algebraic Thinking (SPBR).
These lessons are essential for
students in BE3/4.
Key To Algebra
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Pre-Assessment
Read and Interpret Symbolic
Information
Ex. Show repeated multiplication for
simple whole numbers using
exponents (such as 32 = 9)
Use the Opening the Unit
Initial Assessment in the
SPBR Teacher Book (pp.
161-164)
Practice Tests in Books 1,2,3
Achieving TABE Success M
Pretest Part B: Applied Math,
Problems 3, 24, and 27
Patterns, Functions, and Algebra
Heads up!!!
All Key to Algebra lessons
include Positive and
Negative Integers. See note
below.
Symbol Sense Practice:
Book 2, Variables, Terms,
pp. 144,145
and Expressions, pp. 6,7, and
Students practice recognizing 15
the difference between
squaring and doubling;
students practice representing
repeated multiplication in
different ways.
Symbol Sense Practice:
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
Math Sense: Algebra and
Geometry Powers and Roots,
p. 16
78
Patterns, Functions, and Algebra
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the
lessons below are from Seeking
Patterns, Building Rules:
Algebraic Thinking (SPBR).
These lessons are essential for
students in BE3/4.
Key To Algebra
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Ex. Represent operations using
different symbols (such as
parentheses or • for multiplication and
/ for division)
Ex. Identify, simply describe, and use
common properties of operations
(associative and distributive property)
Ex. Apply order of operations to
evaluate expressions
Ex. Write statements of equality or
inequality (such as 3 > 4 – 3)
pp. 16, 17
Students represent
multiplication and division in
multiple ways.
EMPower Everyday Number
Sense, Lesson 7- Patterns and
Order
Symbol Sense Practice:
pp. 28,29
Students practice evaluating
expressions, inserting
operations signs and
parentheses to make
statements true.
p. 45, Students apply order of
operations to algebraic
expressions.
p. 97, Students evaluate
geometric formulas
Symbol Sense Practice:
p. 44, Students translate
algebraic equations into
words and vice-versa.
pp. 85, 119 Students practice
use of < , > , =, ≠, ≈
Symbol Sense Practice:
Achieving TABE Success M
Reviewing Basic Number
Properties, p. 114-115
Book 1: Operations on
Integers, pp. 28-32
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
Achieving TABE Success M
Evaluating Expressions, p.
119
Achieving TABE Success D
Evaluating Expressions, p.
104
Math Sense: Algebra and
Geometry Order of
Operations, pp. 18-19
Achieving TABE Success M
Using Equations, p. 120
Achieving TABE Success D
Interpreting Expressions and
Equations, pp. 105
79
Patterns, Functions, and Algebra
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the
lessons below are from Seeking
Patterns, Building Rules:
Algebraic Thinking (SPBR).
These lessons are essential for
students in BE3/4.
Key To Algebra
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Ex. Families
Ex. Solve a variety of equations and
inequalities
pp. 61, 72
Symbol Sense Practice:
pp. 107,108, Students solve
one-step equations.
pp. 120-121, 132, Two-step
equations
Book 3: Equations
(with integers)
In Lessons 1-3, students
complete and construct
tables, write rules that
generalize patterns revealed
through the tables, and work
through a problem using
these skills.
Book 8: Graphs
Using positive and negative
integers, the book connects
graphs, tables, equations, but
NOT situations.
Achieving TABE Success M
Using Addition and
Subtraction to Solve
Equations, p. 121
Interpret and apply a wide variety
of common patterns, functions,
and relationships
Ex. Recognize and develop repeating
patterns and generalize the
relationship with a table, rule, graph,
or formula
Ex. Generalize, in words, the
relationship between quantities in a
table of amounts (including in-out
tables)
In Lessons 4-6, students
review graphing conventions,
graph table data, and compare
graphs for positive and
negative linear relationships.
Function Tables
TABE Fundamentals M
http://www.worksheetworks. Functions and Patterns, pp.
com/math/geometry/graphi 42-43
ng/function-table.html
TABE Fundamentals D
The Cake Problem
Functions and Patterns, pp.
TIAN Bundle 4
40-41
pp. 4-5
Missing elements, pp. 42-43
http://adultnumeracy.terc.ed
u/TIAN_bundle4.html
Math Sense: Comprehensive
Review x/y tables
More Problems Like Phone w/equations, pp. 170-173.
Plans
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
80
Patterns, Functions, and Algebra
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the
lessons below are from Seeking
Patterns, Building Rules:
Algebraic Thinking (SPBR).
These lessons are essential for
students in BE3/4.
Key To Algebra
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Ex. Develop formulas and create
simple linear graphs from tables
Ex. Use variables to explain real-life
situations.
Ex. Create and use algebraic
expressions and equations to model
situations and solve problems
TIAN Bundle 4
pp. 11–19
http://adultnumeracy.terc.ed
u/TIAN_bundle4.html
In Lesson 7, students
Examine equations by
translating rules into
equations and connecting
equations with graphs, tables,
rules, and situations.
In Lessons 8-9, students solve
problems using a repertoire
of representations, as they
explore linearity and
constant rate of change.
Achieving TABE Success M
Identifying Geometric
Patterns, pp. 110-111
Identifying Number Patterns,
p. 112
Working with Functions,
p.116
Writing Algebraic
Expressions, pp.117-118
In Lessons 10-12, students
contrast linear and nonlinear
situations to deepen
understanding of constant
and nonconstant rates of
change as represented in
tables, graphs, and equations.
Achieving TABE Success M
Algebra Skills Checkup, pp.
122-123
Review and Test Practice
Post-Assessment
Get It Together, p. 135
Annabelle Arable and p. 136
The Bus Stops Here
Closing the Unit
Practice Tests in Books
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
81
Patterns, Functions, and Algebra
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the
lessons below are from Seeking
Patterns, Building Rules:
Algebraic Thinking (SPBR).
These lessons are essential for
students in BE3/4.
Key To Algebra
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Final Assessment (TB pp.
191-200)
1,2,3,8
Note: All EMPower “Symbol Sense Practices” from Seeking Patterns, Building Rules: Algebraic Thinking incorporate positive integers only. In contrast, all Key to Algebra lessons make use of positive and negative integers. Teachers
should keep these features of the materials in mind when making decisions about integrating these resources. For example, if a pre-assessment reveals that some students in the class have a fragile understanding of working with
negative integers, it will be important to provide instruction in using the 4 basic operations with positive and negative integers before using Key to Algebra lessons with them. (See “Suggested Resources for Developing Number Sense
and Operations…” for suggested lessons, activities)
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
82
Patterns, Functions, and Algebra
GED
Suggested Resources for Developing Patterns, Functions, and Algebra Proficiency at the GED Level
To be sufficiently prepared to pass the GED, manage the mathematical demands of work and family, and to begin preparation for post-secondary readiness, students should develop
proficiency with the following content topics related to Patterns, Functions, and Algebra.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

Pre-Assessment
Read and Interpret Symbolic
Information
Ex. Operations on integers
EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
Use the Final
Assessment in the SPBR
Teacher Book (pp. 191200) to determine
students’ fundamental
understandings in
algebra. Then choose
appropriate lessons
described below to give
students more hands-on
experience, especially in
the patterns and
functions cluster.
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Practice Tests
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Achieving TABE Success D
Pretest Part B: Applied
Math, Problems 5, 20
Book 1: Operations on
Integers, pp. 36-37
Number Power Review
Patterns, Functions, and
Algebra Pretest:
Problems
pp. 35-41, Solving
Equations and Graphing
a Linear Equation
Book 2: Variables,
Terms, and Expressions,
pp. 36-37
Book 3: Equations, pp.
36-37
Heads up!!!
All Key to Algebra lessons
include Positive and
Negative Integers. See
note below.
Book 1: Operations on
Integers
Other Resources OACE
Teachers Currently Use
Lesson 5: The Number
Line and Coordinate
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
TABE Fundamentals
Computation M Adding,
83
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Grid
Ex. Represent operations using
different symbols (such as
parentheses or • for multiplication
and / for division)
Ex. Show repeated multiplication for
simple whole numbers using
exponents (such as 32 = 9)
Symbol Sense Practice:
pp. 16-17
Students represent
multiplication and
division in multiple
ways.
Symbol Sense Practice:
pp. 144-145
Students practice
recognizing the
difference between
squaring and doubling;
students practice
representing repeated
multiplication in
different ways.
Book 2
pp. 6-7, Exponents
p. 15, Finding Powers
with a Calculator
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
subtracting, multiplying
and dividing integers
pp. 38-46
Achieving TABE Success D
Integer Skill checkup pp.
79-80.
Understanding Integers,
p. 73
Working with Absolute
Value, p. 74
Adding, p. 75
Subtracting, p. 76
Multiplying and
Dividing, p. 77
Lesson 11: Exponents
Squares
Square Roots
Lesson 13: More
Powers- Powers of Ten
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
84
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

Ex. Identify, simply describe, and
use common properties of
operations (associative and
distributive property)
Ex. Apply order of operations to
evaluate expressions
EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
EMPower Everyday
Number Sense, Lesson 7Patterns and Order
Symbol Sense Practice:
pp. 28,29
Students practice
evaluating expressions,
inserting operations
signs and parentheses to
make statements true,
p. 45, Students apply
order of operations to
algebraic expressions
p. 97, Students evaluate
geometric formulas
Book 1
pp. 29-32, Order of
Operations
Lesson 11:
Order When Evaluating
Expressions
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Achieving TABE Success D
Reviewing Basic
Properties of Numbers,
pp. 100-101
Math Sense: Algebra and
Geometry
The Distributive
Property, pp. 42-43
Achieving TABE Success D Practice Questions for
Evaluating Expressions, TABE D 9/10 (Steve
p. 104
Meyerson), Recognizing
and/or solving
expressions with
decimals, p. 32
85
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

Ex. Write statements of equality or
inequality (such as 3 > 4 – 3)
EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Symbol Sense Practice:
p. 44, Students translate
algebraic equations into
words and vice-versa.
pp. 85, 119 Students
practice use of < , > , =,
≠, ≈
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Achieving TABE Success D
Interpreting Expressions
and Equations, p. 105.
Writing equations and
inequalities, p. 109
Top 50 Math Skills for
GED Success
pp. 94-101
Math Sense: Comprehensive
Review
Inequalities, pp. 70-71
Ex. Understand and use inverse
operations
Ex. Solve a variety of equations and
inequalities
Symbol Sense Practice:
pp. 61, 72
Symbol Sense Practice:
pp. 107,108, Students solve
one-step equations.
pp. 120-121, 132, Two-step
equations
Book 3: Equations
(with integers)
Lesson 3: Equivalent
Equations: Addition and
Subtraction
Lesson 9: Equivalent
Equations: Multiplication
and Division
Ex. Solve a variety of equations and
inequalities (continued)
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
Get It Together
pp. 38 Number Shapes 3 p.
39 Number Shapers 4
OACE Math Kit: Solving
Algebraic Equations, pp. 2023
Achieving TABE Success D
Solving Equations, pp. 104107
Solving Inequalities, p. 108
Practice Questions for
TABE D 9/10 (Steve
Meyerson), Inequalities, p.
31
Achieving TABE Success M
Solving Equations, p. 121
2002 GED Practice
Problems similar to OPT
PA,PB,PC: Solving
86
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
OACE Math Kit Solving
algebraic equations, pp. 20-23
Math Sense: Comprehensive
Review pp. 70-71
Number Power Review
Solving Addition and
Subtraction Equations, pp.
176-177
Solving Multiplication and
Division Equations, pp. 178179
Solving Multistep Equations,
pp. 180-181
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Equations, pp. 12-14
2007 GED Practice
Problems similar to OPT
PF and PG: Solving
Equations, pp. 12-14
Math Sense: Algebra and
Geometry
Solving Equations, pp. 56-63
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
87
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Interpret and apply a wide
variety of common patterns,
functions, and relationships
Ex. Recognize and develop
repeating patterns and generalize
the relationship with a table, rule,
graph, or formula
Ex. Generalize, in words, the
relationship between quantities in a
table of amounts (including in-out
tables)
In Lessons 1-3, students
Complete and construct
tables, write rules that
generalize patterns
revealed through the
tables, and work through
a problem using these
skills
In Lessons 4-6, students
Review graphing
conventions, graph table
data, and compare
graphs for positive and
negative linear
relationships
In Lesson 7, students
Examine equations by
translating rules into
equations and
Book 8: Graphs
Using positive and
negative integers, the
book connects graphs,
tables, equations, but
NOT situations.
Lesson 24: Relating
Rates and Slopes to
Graphs
Function Tables
Top 50 Math Skills
http://www.worksheetw pp. 102-107
orks.com/math/geomet
ry/graphing/functionAchieving TABE Success
table.html
M
Identifying Geometric
The Cake Problem
Patterns, pp. 110-111
TIAN Bundle 4,
Identifying Number
pp. 4-5
Patterns, p. 112
http://adultnumeracy.te In/Out tables, p. 116
rc.edu/TIAN_bundle4.h
tml
Achieving TABE Success D
Identifying Number
More Problems Like
Patterns, p. 99
Phone Plans
Working with Functions,
TIAN Bundle #4
p. 102
pp. 11–19
Interpreting Expressions
http://adultnumeracy.te and Equations, p.105
rc.edu/TIAN_bundle4.h Using Equations to
tml
Solve Word Problems,
p. 110
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
2002 GED Practice
Problems similar to OPT
PA, PB, PC: Numbers,
Squares, Fibonacci, etc.
p. 22
OACE Math Kit Solving
algebraic equations, pp.
20-23
88
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

Ex. Develop formulas and create
simple linear graphs from tables
EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
connecting equations
with graphs, tables,
rules, and situations
In Lessons 8-9, students
Solve problems using a
repertoire of
representations, as they
explore linearity and is
constant rate of change
Ex. Use variables to explain real-life
situations.
In Lessons 10-12,
students
Contrast linear and
nonlinear situations to
deepen understanding of
constant and nonconstant rates of change
as represented in tables,
graphs, and equations.
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Exploring Signed
Number Models
TIAN Bundle #3 pp. 6–
13
http://adultnumeracy.te
rc.edu/TIAN_bundle3.h
tml
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
TABE Fundamentals
Applied M
Applied: Functions and
Patterns, pp. 42-43
Patterns and Shapes, pp.
72-73
TABE Fundamentals
Applied D
Functions and Patterns,
pp. 40-41
Patterns and Shapes, pp.
72-73
Color Chips –
Subtraction
http://nlvm.usu.edu/en
/nav/frames_asid_162_
g_3_t_1
.html?from=category_g_ Math Sense
3_t_1.html
x/y tables w/equations,
pp. 170-173
Number Power
pp. 209-210
Contemporary’s Pre-Algebra
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
89
Patterns, Functions, and Algebra
GED
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS

EMPower
Unless stated otherwise,
the lessons below are from
Seeking Patterns, Building
Rules: Algebraic Thinking
(SPBR)
Many of the lessons in
SPBR are beneficial for
GED students
The Math Problem
Key To Algebra
Solver
OACE Teacher
Developed Resources
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information
about lessons and activities
teachers have developed
and found to be effective
Becoming familiar with
equations, p. 12
Ex. Create and use algebraic
expressions and equations to model
situations and solve problems
Get It Together
p. 135 Annabelle Arable
and p. 136 The Bus
Stops Here
Achieving TABE Success D
Algebra Skills Checkup,
pp. 111-112
Review and Test Practice
Number Power Review
Algebra Review, pp.
202-203
Post-Assessment
Note: All EMPower “Symbol Sense Practices” from Seeking Patterns, Building Rules: Algebraic Thinking incorporate positive integers only. In contrast, all Key to Algebra lessons make use of positive and negative
integers. Teachers should keep these features of the materials in mind when making decisions about integrating these resources. For example, if a pre-assessment reveals that some students in the class have a
fragile understanding of working with negative integers, it will be important to provide instruction in using the 4 basic operations with positive and negative integers before using Key to Algebra lessons with them.
(See “Suggested Resources for Developing Number Sense and Operations…” for suggested lessons, activities).
OACE AIM Patterns, Functions, and Algebra. Version 1 Fall 2011
90
Geometry and Measurement
BE 1/2
Suggested Resources for Developing Geometry and Measurement Proficiency at the BE1/2 Levels
To be sufficiently prepared to complete the BE1/2 Levels and begin the BE 3 Level, manage the mathematical demands of work and family, students should develop proficiency with the
following content topics related to Geometry and Measurement.
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the lessons
come from EMPower’s Over, Around,
and Within: Geometry and
Measurement (OAW). These lessons
are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for
BE Levels 1/2.
and
CONTENT TOPICS
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Pre-Assessment
Begin with the Initial Assessment
in the OAW Teacher Book (pp.
161-164) to determine students’
fundamental understandings of
geometry and measurement. Then
choose appropriate lessons
described below to give students
more hands-on experience.
Read, write, interpret, and apply very
simple types of information related to
measurement and geometry
and
Recall and use a few simple mathematical
procedures
Measurement
Ex. Read and write time (hours and minutes,
day and months, etc.)
Achieving TABE Success E
Choosing the best tool, p. 107
Reading a scale, pp. 108-109
Estimating a measurement on a
OACE AIM Geometry and Measurement. Version 1 Fall 2011
91
Geometry and Measurement
BE 1/2
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the lessons
come from EMPower’s Over, Around,
and Within: Geometry and
Measurement (OAW). These lessons
are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for
BE Levels 1/2.
and
CONTENT TOPICS
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

scale, p. 110
Measuring temperature, p. 111
Measuring length, p. 112
Estimating length, p. 113-116
Measuring weight and mass, p. 117
Measuring liquids, p. 118
Reading Time, p. 119
Calculating Time, p. 120
Finding Elapsed Time, p. 121
Ex. Show equivalent amounts of money using
different coins and dollar bills
Ex. Read and interpret common measuring
tools (rulers, measuring cups, gauges, scales,
meters, etc.) to nearest whole unit
Ex. Develop benchmark estimates for common
measures (such as inch, foot, yard, pound)
TABE Fundamentals Applied Math M
Appropriate Instrument, pp. 50-51
Time. pp. 54-55
Ex. Recognize that a given measurement
consists of more small units than large units
(i.e., it takes 36 inches to make a yard, but only
3 feet)
Ex. Identify and select appropriate units of
metric and customary measures
Ex. Use appropriate tools to measure highfrequency standard whole units of measure
(i.e., pounds, inches, feet, gallons, cups) to the
nearest whole unit
Lines, Angles, and Shapes
Ex. Find lengths
Ex. Use spatial visualization skills (e.g., rotating
OAW (with adaptation)
In Lessons 1, 2 & 3, students:
 Use language such as “sides,”
“parallel,” and “perpendicular”
to describe and name shapes
Math Sense: Measurement and Data
Analysis
Using Units of Time, pp. 40-43
English Units of Capacity, pp. 5255
Metric Units of capacity, pp. 56-59
English Units of Weight, pp. 66-69
Metric Units of Weight, pp. 70-73
Measuring Temperature, pp. 78-79
Achieving TABE Success M
Distinguishing among
transformations, p. 124
OACE AIM Geometry and Measurement. Version 1 Fall 2011
92
Geometry and Measurement
BE 1/2
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the lessons
come from EMPower’s Over, Around,
and Within: Geometry and
Measurement (OAW). These lessons
are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for
BE Levels 1/2.
and
CONTENT TOPICS
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

and translating shapes; recognizing benchmark
fractions




and their attributes.
Count sides and angles to
compare and contrast shapes.
Establish a working definition
for “angle.”
Use right (90º) and straight
(180º) angles as benchmarks to
estimate angle size.
Measure angles with a
protractor.
Interim Assessment 1: Shapes and
Angles, pp. 169-171
2-Dimensional Shapes (Area and Perimeter)
Polygons
Ex. Identify shapes (and solids)
Ex. Find the perimeter and area of rectangles
(and other 2-D shapes) using simple strategies
Ex. Combine and separate figures and describe
their geometric attributes and properties (i.e., a
rectangle can be divided into two right triangles)
Get It Together, Polygons (Area)
Polygon 1, p. 63
Polygon 2, p. 64
Achieving TABE Success E
Identify Plane Figures, pp. 125-127
Visualizing shapes, pp. 128-129
Recognizing Congruent shapes, p.
130
Recognizing similar shapes, p. 131
Identifying line of symmetry, p. 132
TABE Fundamentals Applied Math M
Plane Figures, pp. 66-67
Achieving TABE Success M
OACE AIM Geometry and Measurement. Version 1 Fall 2011
93
Geometry and Measurement
BE 1/2
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the lessons
come from EMPower’s Over, Around,
and Within: Geometry and
Measurement (OAW). These lessons
are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for
BE Levels 1/2.
and
CONTENT TOPICS
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Coordinate plane
Locating points on the coordinate
plane, p. 126
Ex. Apply simple coordinate graphing elements
(positioning of points on a scale or number line
and rectangular coordinates) to specify
locations and relationships (such as east and
west, north and south on a map)
Review and Test Practice
Post-Assessment
OACE AIM Geometry and Measurement. Version 1 Fall 2011
94
Geometry and Measurement
BE 3/4
Suggested Resources for Developing Geometry and Measurement Proficiency at the BE3/4 Levels
To be sufficiently prepared to complete the BE3/4 Levels and begin the GED level, manage the mathematical demands of work and family, and to anticipate preparation for post-secondary
readiness, students should develop proficiency with the following content topics related to Geometry and Measurement.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

Pre-Assessment
Begin with the Initial Assessment in
the OAW Teacher Book (pp. 161164) to determine students’
fundamental understandings of
geometry and measurement. Then
choose appropriate lessons described
below to give students more handson experience.
Read, write, interpret, and apply
very simple types of information
related to measurement and
geometry
and
Recall and use a good store of
mathematical procedures
In Lessons 8 & 9, students:
 Convert between standard
English linear and square
Ex. Recognize and use commonly
units (metric units are used
used standard units of measurement to
Measurement
Achieving TABE Success M
Understanding Measurement, p. 124
Choosing Appropriate Unit of
Measure, p. 125
OACE AIM Geometry and Measurement. Version 1 Fall 2011
95
Geometry and Measurement
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

the nearest half and quarter
Ex. Identify and select appropriate units of
metric and customary measures
in the previous OAW Units
5-7)
Ex. Use appropriate tools to measure
to the degree most appropriate (e.g.,
nearest half or quarter unit) for the
situation
Ex. Make simple conversions within
the same measurement system (such
as inches to feet, or centimeters to
meters)
Ex. Convert units of measure from one
system to another using informal
methods (i.e., a centimeter is about
half an inch)
Ex. Add and subtract dimensional
numbers (such as 5’2” – 3’7”, or 5:20
AM minus 3:40 AM) to solve problems
involving measurement and geometry
OAW (with adaptation)
Lines, Angles, and Shapes
In Lessons 1, 2 & 3, students:
 Use language such as
Ex. Recognize and describe two“sides,” “parallel,” and
dimensional shapes, including basic
“perpendicular” to describe
angle descriptions (such as acute, right
Converting Units within the
Customary System, pp. 126-127
Converting Units within the Metric
System, pp. 128-129
Adding and Subtracting Units of
Measure, p. 130
Calculating Time, p. 138
Finding Elapsed Time, p. 139
Adding and Subtracting Time, p. 140
Achieving TABE Success D
Reviewing Customary Units of
Measure, p. 133
Converting Units within the
Customary System, p. 134
Reviewing Metric Units of Measure,
p. 135
Converting Units within the Metric
System, p. 136
Adding and Subtracting Mixed
Measurements, p. 138
Calculating Time, p. 139
Finding Elapsed Time, p. 140
Triangles in Real Life
Get It Together
TIAN Bundle 5, pp. 4–8
Geometrical Constructions (Stick
http://adultnumeracy.terc.edu/TIAN Figures)
_bundle5.html
Stick Figures 1, p. 51
Stick Figures 2, p. 52
OACE AIM Geometry and Measurement. Version 1 Fall 2011
96
Geometry and Measurement
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

and obtuse)
Ex. Use benchmark angles (such as
90 degrees and 45 degrees) to
estimate size of angles


Ex. Recognize and describe parallel,
perpendicular and skew lines

Ex. Spatial visualization of symmetry,
parallelism, perpendicularity, etc.

and name shapes and their
attributes.
Count sides and angles to
compare and contrast
shapes.
Establish a working
definition for “angle.”
Use right (90º) and straight
(180º) angles as benchmarks
to estimate angle size.
Measure angles with a
protractor.
Interim Assessment 1: Shapes and
Angles, pp. 169-171
Stick Figures 3, p. 53
Stick Figures 4, p. 54
Pattern Blocks:
Left, Right, Middle, End, p. 57
Oh Hexagon, p. 58
Terry’s Triangle, p. 59
Glenda’s Pattern, p. 60
Polygons (Area):
Polygon 3, p. 65
Polygon 4, p. 66
Polygon 5, p. 67
Constructions (Length of Segment):
Length of AB, p. 127
Achieving TABE Success M
Naming Angles, p. 144
Investigating Lines, p. 145
Achieving TABE Success D
Recognizing types of angles, p. 114
Recognizing relationships of lines, p.
115
Recognizing types of triangles, p. 118
OACE AIM Geometry and Measurement. Version 1 Fall 2011
97
Geometry and Measurement
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

2-Dimensional Shapes (Area and
Perimeter)
Polygons
Ex. Solve and estimate solutions to
problems involving length, perimeter,
area of common two-dimensional
shapes
Ex. Describe the difference between
square units and linear units and when
each is used
In Lessons 5, 6 & 7, students:
 Distinguish between area and
perimeter
 Derive the formulas for area
and perimeter
In Lessons 8 & 9, students:
Convert between standard English
linear and square units (metric units
are used in the previous OAW Units
5-7)
Interim Assessment 2: A Fresh Look,
pp. 173-180
Seeking Patterns, Building Rules: Algebraic
2-Dimensional Shapes (Area and
Thinking
Perimeter)
In Lesson 6: Circle Patterns, students:
 Describe the approximate
Circles
relationship between the
diameter and circumference of
Ex. Measure and compare radius,
diameter, and circumference of a circle
circle
and informally develop a rule and an
equation for determining the diameter
or circumference
3-Dimensional Shapes (Volume and
surface area)
Don't Fence Me In!
Achieving TABE Success M
TIAN Bundle 5, pp. 12–19
Finding Perimeter, p. 135
http://adultnumeracy.terc.edu/TIAN Finding Area, p. 136
_bundle5.html
Achieving TABE Success D
Parallelogram Activity
Identifying Polygons, pp. 116-117
NCTM Illuminations website
Finding Perimeter, p. 141
http://illuminations.nctm.org/Activit Finding the Area of Squares and
yDetail.aspx?ID=108
Rectangles, p. 143
Finding the Area of a Triangle, pp.
144-145
Achieving TABE Success M
Investigating Circles, p. 149
Achieving TABE Success D
Identifying parts of a circle, p. 120
Finding Circumference, p. 142
Finding the Area of a Circle, p. 146
Number Power: Geometry
Working with Circles, pp. 110-113
In Lessons 11, 12 & 13, students:
 Determine the capacity of
rectangular solids
Cube Nets Activity
Get It Together
NCTM Illuminations website
Build It! (Build a small structure out
http://illuminations.nctm.org/Activit of cubes)
OACE AIM Geometry and Measurement. Version 1 Fall 2011
98
Geometry and Measurement
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

Ex. Recognize and describe 3-D solids
Ex. Draw on understanding of area to
develop and justify the formula for
volume of a cube or rectangular solid
 Contrast surface area and
volume
In Lessons 4 & 10, students:
Similarity, scale, and applications of
 Use rulers to find perimeter
proportional reasoning
for, enlarge, and measure twodimensional shapes
Ex. Use direction, distance, labels,
 Distinguish between similar
simple scales, and symbols to read
and use maps and plans
and non- similar shapes
 Make scale drawings
Ex. Determine whether twodimensional shapes have similar
attributes and properties (i.e., are they
congruent?)
yDetail.aspx?ID=84
Build It
Build It
Build It
Build It
#1, p. 45
#2, p. 46
#3, p. 47
#4, p. 48
Achieving TABE Success D
Working with 3-D figures, pp. 122123
Finding Surface Area, p. 147
Finding Volume, p. 148
Achieving TABE Success M
Recognizing Congruent Figures, p.
152
Recognizing Similar Figures, p. 153
Comparing Three-Dimensional
Figures, p. 154
Reasoning With Ratios: Keeping
Things in Proportion
In Lesson 6: Redesigning Your
Calculator, students:
 Measure and draw similar
shapes
Review and Test Practice
OACE AIM Geometry and Measurement. Version 1 Fall 2011
99
Geometry and Measurement
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Over, Around, and
Within: Geometry and Measurement
(OAW). These lessons are aimed at
Levels 3 and 4 students.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons and
activities teachers have developed and
found to be effective

Post-Assessment
Closing the Unit (Design a Box, Mind
Map)
Final Assessment pp. 181-187
OACE AIM Geometry and Measurement. Version 1 Fall 2011
100
Geometry and Measurement
GED
Suggested Resources for Developing Geometry and Measurement Proficiency at the GED Level
To be sufficiently prepared to pass the GED, manage the mathematical demands of work and family, and to begin preparation for post-secondary readiness, students should develop
proficiency with the following content topics related to Geometry and Measurement.
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Opening the Unit (see OACE
class example for Geometry
Mind Map in a GED class)
Pre-Assessment
Number Power: Geometry
Geometry Pretest, p. 16
Use the Final Assessment in
the OAW Teacher Book (pp.
181-187) to determine
students’ fundamental
understandings of each
geometry and measurement
content topic. Then choose
appropriate lessons described
below to give students more
hands-on experience.
OACE AIM Geometry and Measurement. Version 1 Fall 2011
101
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

Read, write, interpret, and apply
very simple types of information
related to measurement and
geometry
and
Recall and use a good store of
mathematical procedures
In Lessons 1, 2 & 3, students: Lesson 4: Geometry Topics
 Use language such as
“sides,” “parallel,” and
Ex. Measure size of angles using tools
“perpendicular” to
such as protractors
describe and name
shapes and their
Ex. Solve problems involving
perpendicularity and parallelism
attributes.
 Count sides and angles
Ex. Apply the knowledge of properties
to compare and contrast
of triangles (e.g., 180 in a triangle) to
shapes.
solve problems
 Establish a working
Ex. Visualizing translations, symmetry
definition for “angle.”
 Use right (90º) and
straight (180º) angles as
benchmarks to estimate
angle size.
 Measure angles with a
protractor.
Angles and shapes
Triangles in Real Life
TIAN Bundle 5, pp. 4–8
http://adultnumeracy.terc.ed
u/TIAN_bundle5.html
OACE AIM Geometry and Measurement. Version 1 Fall 2011
Get It Together
Geometrical Constructions
(Stick Figures):
Stick Figures 1, p. 51
Stick Figures 2, p. 52
Stick Figures 3, p. 53
Stick Figures 4, p. 54
Pattern Blocks:
Left, Right, Middle, End, p.
57
Oh Hexagon, p. 58
Terry’s Triangle, p. 59
Glenda’s Pattern, p. 60
Polygons (Area):
Polygon 3, p. 65
Polygon 4, p. 66
Polygon 5, p. 67
Constructions (Length of
Segment):
102
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

OAW Interim Assessment 1:
Shapes and Angles, pp. 169171
Length of AB, p. 127
Achieving TABE Success D
Visualizing among
transformations, pp. 124-125
Achieving TABE Success M
Working with Triangles, p.
148
Recognizing Symmetry, p.
151
In Lessons 5, 6 & 7, students:
Don't Fence Me In!
Number Power: Geometry
2-Dimensional Shapes (Area and
Distinguish between area and Lesson 8: Measurement Units TIAN Bundle 5, pp. 12–19
Perimeter, pp. 88-89
Perimeter)
perimeter
and Polygons, pp. 77-83
http://adultnumeracy.terc.ed Recognizing Common
Derive the formulas for area
u/TIAN_bundle5.html
Polygons, pp. 90-91
Polygons
and perimeter
Area, pp. 92-93
Ex. Solve and estimate solutions to
Parallelogram Activity
Working with Squares, pp.
problems involving length, perimeter,
NCTM Illuminations website 94-97
area of common two-dimensional shapes In Lessons 8 & 9, students:
Convert between standard
http://illuminations.nctm.org Working with Rectangles, pp.
Ex. Develop formulas for finding area of
English linear and square
/ActivityDetail.aspx?ID=108 98-101
different polygons and explain their
units (metric units are used in
Working with Triangles, pp.
relationship
the previous OAW Units 5-7)
102-105
Working with Parallelograms
Ex. Predict the impact of changes in
Interim Assessment 2: A
and Trapezoids, pp. 106-109
linear dimensions on the perimeter and
Fresh Look, pp. 173-180
Perimeter and Area:
area of a polygon
Applying your Skills, pp. 114Ex. Determine the dimensions of various
124
common two-dimensional shapes (i.e.,
length when the area and width are
OACE AIM Geometry and Measurement. Version 1 Fall 2011
103
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

given)
Ex. Use the language of metric unit
prefixes to describe real-life
measurements
2-Dimensional Shapes (Area and
Perimeter)
Circles
Ex. Determine and use the
relationship between the diameter and
the circumference of a circle and the
area of the circle
3-Dimensional Shapes (Volume and
surface area)
Ex. Develop and justify formulas for
prisms and cylinders
Ex. Predict the impact of changes in
linear dimensions on the perimeter,
Seeking Patterns, Building Rules: Lesson 12: Circles
Algebraic Thinking
In Lesson 6: Circle Patterns,
students:
 Describe the
approximate relationship
between the diameter
and circumference of
circle
In Lessons 11, 12 & 13,
Lesson 8: 2 and 3D, pp. 86students:
87
 Determine the capacity
of rectangular solids
 Contrast surface area
and volume
Number Power: Geometry
Working with Circles, pp.
110-113
Cube Nets Activity
NCTM Illuminations website
http://illuminations.nctm.org
/ActivityDetail.aspx?ID=84
Get It Together
Build It! (Build a small
structure out of cubes)
Build It #1, p. 45
Build It #2, p. 46
Build It #3, p. 47
Build It #4, p. 48
Closing the Unit: Design a
OACE AIM Geometry and Measurement. Version 1 Fall 2011
104
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

area, and volume and surface area of
a figure
Box
Number Power: Geometry
Recognizing Common Solid
Figures, pp. 127-128
Volume, pp. 129-130
Working with Cubes, pp.
131-132
Working with Rectangular
Solids, pp. 133-134
Working with Cylinders, p.
135
Working with Cones, p. 136
Volume: Applying your Skills,
pp. 137-145
Ex. Determine the dimensions of
various common three-dimensional
shapes (i.e., length when the area is
given, or the width if the height, length,
and volume are given)
In Lessons 4 & 10, students:
 Use rulers to find
perimeter for, enlarge,
and measure twoEx. Use ration and proportion to solve
dimensional shapes
problems involving scale drawings or
similar figures
 Distinguish between
similar and non- similar
Ex. Solve problems involving
shapes
congruence and similarity of geometric
 Make scale drawings
figures
Lesson 21: pp. 220-222
Similarity, scale, and applications
of proportional reasoning
Lesson 11: pp. 119-125
Pythagorean Theorem
Ex. Use Pythagorean Theorem to
OACE AIM Geometry and Measurement. Version 1 Fall 2011
105
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

solve problems involving right triangles
Coordinate plane
Ex. Use coordinates to describe
geometric figures or their
transformations (e.g., rotations,
reflections)
Linear functions and graphs
Lesson 24 (see Algebra
covered extensively in Seeking resources)
Patterns, Building Rules:
Algebraic Thinking (see Algebra
resources)
Ex. Determine the slope of a line, the
y-intercept, and the intersection of two
lines
Achieving TABE Success M
Calculating Time, p. 138
Finding Elapsed Time, p. 139
Adding and Subtracting
Time, p. 140
Measurement
Ex. Use uniform rates (e.g., miles per
hour, bushels per acre) in problem
situations
Ex. Add and subtract dimensional
numbers (such as 5’ 2” - 3’ 7”, or such
as 5:20 AM minus 3:40 AM) to solve
problems involving measurement and
geometry
Review and Test Practice
Post-Assessment
Final Assessment pp. 181-187
to assess fundamental
Taken together, the following
pages assess readiness to
OACE AIM Geometry and Measurement. Version 1 Fall 2011
106
Geometry and Measurement
GED
EQUIPPED FOR THE FUTURE
EMPower
STANDARD
Unless stated otherwise, the
lessons come from EMPower’s
Over, Around, and Within:
Geometry and Measurement
(OAW). Many of the lessons in
OAW are beneficial for GED
students.
and
CONTENT TOPICS
OACE Teacher Developed
Resources
The Math Problem Solver
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective

concepts
handle many of the geometry
and measurement demands of
the GED: pp. 46,47,86,87,
124,125,132,133
OACE AIM Geometry and Measurement. Version 1 Fall 2011
107
Data, Statistics, and Probability
BE 1/2
Suggested Resources for Developing Data, Statistics, and Probability Proficiency at the BE1/2 Levels
To be sufficiently prepared to complete the BE1/2 Levels and begin the BE 3 Level, manage the mathematical demands of work and family, students should develop proficiency with the
following content topics related to Data, Statistics, and Probability.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Many Points
Make a Point: Data and Graphs (MPMP).
These lessons are aimed at Levels 3 and
4 students, but can be adjusted in many
cases for BE Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Pre-Assessment
Begin with the Initial Assessment in
the MPMP Teacher Book (pp. 155163) to determine students’
fundamental understandings of data
and statistics. Then choose
appropriate lessons described below
to give students more hands-on
experience.
Achieving TABE Success E Pretest
Part B: Applied Math: Probability –
Problems 13, 14, and 18.
Average – Problems 11 and 15.
Number Power: Analyzing Data
Analyzing Data Pretest, pp. 1-6
Collect, Organize, and Display Data
and
Read and interpret data and
statistical information
It is important for students to have
opportunities to both create data displays as
well as to interpret them. Materials which
provide opportunities to CREATE data
displays are noted with *Create*. Otherwise,
resources are designed for interpretation.
*Create*
*Create*
Achieving TABE Success E Reading a
Creative Examples
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
108
Data, Statistics, and Probability
BE 1/2
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Many Points
Make a Point: Data and Graphs (MPMP).
These lessons are aimed at Levels 3 and
4 students, but can be adjusted in many
cases for BE Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Ex. Use simple check sheets, picture
graphs, and frequency graphs to collect and
organize data based on posed questions
Interpretive Examples
Ex. Extract discrete information from
simple bar graphs, from simple circle
graphs, or from a list or table
Ex. Draw conclusions from information
displayed in simple bar graphs, from
simple circle graphs, or from a list or
table
Ex. Make relative comparisons about
values on a bar graph or other graphic
representation of data (such as “greater
than” or “less than” or “about twice as
much” or “between 30 and 40”)
Ex. Describe concrete information found
in a graphic representation
In Lessons 1 and 2, students:
 Experience the process of
collecting, organizing, and
describing data.
 Use a frequency graph as a
quick way to tally data.
 Generate statements using
common benchmark fractions
and percents to describe data
pictured in frequency graphs.
 Regroup data and discuss the
implications of the change.
 Speculate about a possible
audience or purpose for the
data.
*Create*
In Lessons 3 and 4, students:
 Transform frequency graphs
into bar and circle graphs.
 Comment on the similarities
and differences among
formats.
 Sketch circle graphs based
on division of a circle and
estimates of percents.
 Use benchmark fractions
Create a graph at:
http://nces.ed.gov/nceskids/createa
graph/default.aspx
Table, pp. 80-81
Using a Table, p. 82
Comparing Data, p. 83
Using a Calendar, p. 84
Using a Price List, p. 85
Recognizing Types of Graphs, p. 86
Reading a Pictograph, p. 87
Reading a Bar Graph, p. 88
Reading a Circle graph, p. 89
Math Sense: Measurement and Data
Analysis
Using a Tally Sheet, pp. 90-91
Displaying Data on a Line Plot, pp.
92-93
Sorting a List of Data, pp. 96-97
Reading a Table, p. 106
Completing a Table, p. 109
Reading a Pictograph, pp. 120-121
Reading a Bar Graph, pp. 124-125
Reading a Circle Graph, pp. 138-139
Graphing Family Expenses, pp. 140141
Number Power: Analyzing Data
Organizing Data, Tables and Charts,
pp. 28-31
Bar Graphs, pp. 32-35
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
109
Data, Statistics, and Probability
BE 1/2
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Many Points
Make a Point: Data and Graphs (MPMP).
These lessons are aimed at Levels 3 and
4 students, but can be adjusted in many
cases for BE Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

and percents to describe data
pictured in circle graphs.
Double Bar Graphs, pp. 36-37
Line graphs, pp.38-42
Pictographs, pp. 44-45
Circle Graphs, pp. 46-48
Line Plots, pp. 49-51
Scatter Diagrams, pp. 52-53
The Purpose of a Graph or Chart,
pp. 62-66
Reading Data, pp. 67-72
Seeing Trends/Making Predictions,
p. 86-89
Use probability
Ex. Describe the probability of an event
within the range of “likely”, “possible”,
“unlikely” or “impossible”.
Ex. Find the probability of a single
outcome in a coin toss
Get It Together
Spinners (Probability on Spinners)
A-H Spinners, p. 102
Spinners JR, p. 103
Achieving TABE Success E Exploring
Probability, p. 91
Probability as a Fraction, p. 92
Math Sense: Measurement and Data
Analysis
Introducing Probability, pp. 166-167
Using Probability for Prediction, pp.
168-169
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
110
Data, Statistics, and Probability
BE 1/2
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
come from EMPower’s Many Points
Make a Point: Data and Graphs (MPMP).
These lessons are aimed at Levels 3 and
4 students, but can be adjusted in many
cases for BE Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Achieving TABE Success E Finding an
Average, p. 90
Average
Review and Test Practice
Post-Assessment
Achieving TABE Success E Data and
Probability Skills Checkup, pp. 92-93
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
111
Data, Statistics, and Probability
BE 3/4
Suggested Resources for Developing Data, Statistics, and Probability Proficiency at the BE3/4 Levels
To be sufficiently prepared to complete the BE 3/4 Levels and begin the GED level, manage the mathematical demands of work and family, and to anticipate preparation for post-secondary
readiness, students should develop proficiency with the following content topics related to Data, Statistics and Probability.
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Pre-Assessment
Begin with Opening the Unit
activities, including Initial
Assessment in MPMP Teacher Book
(pp. 141-144) to determine students’
fundamental understandings of data
and statistics.
Achieving TABE Success M Pretest
Part B: Applied Math: Data –
Problems 5, 6, and 7
Probability – Problem 12
Average – Problem 15
Number Power: Graphs, Charts,
Schedules, and Maps
Graph Skills Inventory, pp. 12-15
Schedule and Chart Skills Inventory,
pp. 77-79
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
112
Data, Statistics, and Probability
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Collect, Organize, and Display Data and
Read and interpret data and statistical
information
It is important for students to have
opportunities to both create data displays as
well as to interpret them. Materials which
provide opportunities to CREATE data displays
are noted with *Create*. Otherwise, resources
are designed for interpretation.
*Create*
In Lessons 1 and 2, students:
 Experience the process of
Ex. Collect and organize categorical data
collecting, organizing, and
based on posed question and organize in a
describing data.
bar graph, or line plot
 Use a frequency graph as a
Ex. Collect and organize data based on
quick way to tally data.
posed question and represent the
 Generate statements using
information in a circle graph or stem-andcommon benchmark fractions
leaf plot
and percents to describe data
pictured in frequency graphs.
Ex. Convert bar graphs into circle graphs
and describe how the two are related and
 Regroup data and discuss the
what each graph represents
implications of the change.
 Speculate about a possible
Ex. Understand that scale, sample size and
audience or purpose for the
organization of data can distort
data.
interpretations of data
Creative Examples
*Create*
Create a graph at:
http://nces.ed.gov/nceskids/createa
graph/default.aspx
Achieving TABE Success M Reading a
Table, pp. 96-97
Using Numbers in a Table, p. 98
Using a Price List, p. 99
Working with Graphs, p. 102
Reading a Circle Graph, p. 103
Reading a Bar Graph, p. 104
Reading a Double Bar Graph, p. 105
Reading a Line Graph, p. 106
Visual Literacy Tables and Graphs
Reading Tables, pp. 3-12
Reading Bar Graphs, pp. 13-24
Reading Line Graphs, pp. 25-34
Reading Circle Graphs, pp. 35-42
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
113
Data, Statistics, and Probability
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

*Create*
In Lessons 3 and 4, students:
 Transform frequency graphs
Ex. Extract discrete information from a list,
into bar and circle graphs.
table, bar graph, pictograph, or line plot
 Comment on the similarities
Ex. Make simple comparisons to support or
and differences among
refute statements based on
formats.
representations of data, including both
 Sketch circle graphs based
absolute and relative comparisons
on division of a circle and
Ex. Make predictions or draw conclusions
estimates of percents.
from information displayed in a list, table,
 Use benchmark fractions
bar graph, circle graph, pictograph, or line
and percents to describe
plot
data pictured in circle
graphs.
Ex. Make statements and numerical
Interpretive Examples
comparisons about relative values on a bar
or circle graph (such as “one category is
three times greater than another” or “this
bar extends more than halfway between 25
and 50”)
*Create*
In Lessons 5,6, 8, and 10, students
turn to numerical data and
representations of change over time.
Students’ skills quickly become more
sophisticated as they handle graphs
with two y-axes (two types of
information) and compare graphs
with different scales.
Students will learn to:
 Sketch lines to match
Math Sense: Measurement and Data
Analysis
Comparing Sets of Data, pp. 94-95
Reading a line Graph, pp. 132-133
Drawing a Curve Graph, pp. 134135
Using two Data Sources, pp. 136137
Data in Different Forms, pp. 142143
Number Power: Graphs, Charts,
Schedules, and Maps
Pictographs,
pp. 20-31
Circle Graphs, pp. 32-43
Bar Graphs, pp. 44-55
Line Graphs, pp.56-67
Schedules and Charts, pp. 84-95
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
114
Data, Statistics, and Probability
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective





statements describing
change over time
Precisely describe upward
and downward trend and
periods of stability.
Match graphs and
descriptions of climate data
for five mystery cities
Use tables
Plot points
Use Measures of Central Tendency and
Other Statistical Measures
Ex. Identify, find and/or use the shape,
range, median, mean, and mode of data
Ex. Describe the effect of range and
outliers on median and mean
*Create*
In Lessons 7 and 9, students will
learn to:
 Develop strategies for finding
the median and mean
 Use mean and median to
describe a data set
Achieving TABE Success M Finding
Mean, Median, and Mode, pp. 100101
Math Sense: Measurement and Data
Analysis
Finding the Mean, p. 100
Finding the Median and Mode, pp.
102-103
Choosing a Central Tendency, p.
104
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
115
Data, Statistics, and Probability
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Use Probability
Get It Together
Spinners (Probability on Spinners)
Which Spinner #1, p.104
Which Spinner #2, p. 105
Draw the Spinner #1, p. 107
Spinner #2, p. 108
Ex. Describe the probability of an event
within the range of “likely”, “possible”,
“unlikely” or “impossible”
Ex. Find the probability of a single outcome
in a simple concrete situation with a very
limited number of possible outcomes (i.e.,
the role of a die)
Achieving TABE Success M
Understanding Probability, p. 107
Ex. Find the probability of a single outcome
in a concrete situation and state it as a
ratio, fraction and percent
Review and Test Practice
MPMP Closing the Unit: Stock Picks
Post-Assessment
Achieving TABE Success M Data
Interpretation Skills Checkup, pp.
108-109
Visual Literacy Tables and Graphs
Mixed Practice, pp. 43-45
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
116
Data, Statistics, and Probability
BE 3/4
EQUIPPED FOR THE FUTURE
STANDARD
and
CONTENT TOPICS
EMPower
Unless stated otherwise, the lessons
below are from Many Points Make a
Point: Data and Graphs (MPMP). These
lessons are aimed at levels 3 and 4
students.
OACE Teacher Developed
Resources
Electronic Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective

Number Power: Graphs, Charts,
Schedules, and Maps
Graph Review, pp. 68-75
Schedule and Chart Review, pp. 96101
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
117
Data, Statistics, and Probability
GED
Suggested Resources for Developing Data, Statistics, and Probability Proficiency at the GED Level
To be sufficiently prepared to pass the GED, manage the mathematical demands of work and family, and to begin preparation for post-secondary readiness, students should develop
proficiency with the following content topics related to Data, Statistics, and Probability.
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

Pre-Assessment
OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
Use the Final Assessment in
the MPMP Teacher Book (pp.
155-163) to determine
students’ fundamental
understandings of data and
statistics. Then choose
appropriate lessons described
below to give students more
hands-on experience.
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Achieving TABE Success D
Pretest Part B: Applied Math:
Data Analysis – Problems 1,
2, 3, and 4. Probability –
Problem 25.
Average – Problem 15
Number Power: Graphs, Charts,
Schedules, and Maps
Graph Skills Inventory, pp.
12-15
Schedule and Chart Skills
Inventory, pp. 77-79
118
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
p. 10 Interpret bar graphs
(whole numbers)
p. 20 *Create* bar graphs
(whole numbers)
p. 140 *Create* bar graphs
(whole numbers and
exponents)
p. 173 Interpret tables
(fractions)
p. 178 Interpret Circle graphs
(fractions)
p. 187 *Create* and Interpret
tables (fractions and decimals)
p. 219 Interpret tables (whole
numbers/proportion)
p. 236 Interpret circle graphs
(percent/degrees)
p. 251 Interpret tables
*Create*
Create a graph at:
http://nces.ed.gov/nceskids/
createagraph/default.aspx
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Collect, Organize, and Display Data
and
Read and interpret data and
statistical information
It is important for students to have
opportunities to both create data displays as
well as to interpret them. Materials which
provide opportunities to CREATE data
displays are noted with *Create*. Otherwise,
resources are designed for interpretation.
Creative Examples
Ex. Collect and organize data based on
posed questions and choose appropriate
representations to display the data
collected, depending on the purpose
Ex. Create and interpret histograms to
compare a large set of data and explain
how the size of the categories influences
the shape and interpretation of the graph
Ex. Create scatterplots to compare two
variables and informally estimate lines of
best fit to test hypotheses
Ex. Use the line of best fit to make
predictions and draw conclusions about
data
*Create*
In Lessons 1 and 2, students:
 Experience the process of
collecting, organizing, and
describing data.
 Use a frequency graph as
a quick way to tally data.
 Generate statements using
common benchmark
fractions and percents to
describe data pictured in
frequency graphs.
 Regroup data and discuss
the implications of the
change.
 Speculate about a possible
audience or purpose for
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
Achieving TABE Success D
Gathering Data for a Survey,
p. 153
Reading a Table, p. 154
Using a Price List, p. 155
Using Tables to Make
Comparisons, p. 156
Interpreting Information from
Tables, p. 157
Reading a Circle Graph, p. 159
Finding Numbers
Represented in Circle Graph,
p. 160
Reading a Bar Graph, p. 161
Understanding Data in Bar
Graphs, p. 162
Reading a Line Graph, p. 163
119
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

the data.
*Create*
In Lessons 3 and 4, students:
 Transform frequency
Ex. Extract and/or use information from a
graphs into bar and circle
list, table, chart, or graph
graphs.
Ex. Make predictions, draw inferences, or
 Comment on the
draw conclusions from information
similarities and
displayed in a table, chart, or graph
differences among
formats.
Ex. Make statements to support or refute
an argument using interpretations from
 Sketch circle graphs
data
based on division of a
circle and estimates of
percents.
 Use benchmark fractions
and percents to describe
data pictured in circle
graphs.
Interpretive Examples
OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
(percent)
p. 258 Interpret line graph
prediction (whole numbers)
*Create*
In Lessons 5,6, 8, and 10,
students turn to numerical
data and representations of
change over time. Students’
skills quickly become more
sophisticated as they handle
graphs with two y-axes (two
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Math Sense: Measurement and
Data Analysis
Comparing Sets of Data, pp.
94-95
Reading a line Graph, pp. 132133
Drawing a Curve Graph, pp.
134-135
Using two Data Sources, pp.
136-137
Data in Different Forms, pp.
142-143
Gridding in Data Analysis
Answers, pp. 144-145
Interpolation and
Extrapolation, pp. 150-151
Drawing Conclusions from
Data, pp. 152-153
What more do I need to
Know? (Not enough
information), pp. 154-155
Understanding Correlation,
pp. 158-159
Analyzing Correlation, pp.
160-161
Top 50 Math Skills for GED
Success
Interpolate and Extrapolate,
pp. 74-75
120
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
types of information) and
compare graphs with different
scales.
Students will learn to:
 Sketch lines to match
statements describing
change over time
 Precisely describe upward
and downward trends
and periods of stability
 Match graphs and
descriptions of climate
data for five mystery
cities
 Use tables
 Plot points
More Advanced Interpretation
Ex. Infer meaning from gaps, clusters,
trends, and comparisons of data
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Interpret a Single Line Graph,
pp. 78-79
Understanding a Multiple line
Graph, pp. 80-81
Interpret a Bar Graph, pp. 8283
Understand a Circle Graph,
pp. 84-85
Interpret a Circle Graph, pp.
86-87
Use two Sources of Data, pp.
88-89
Number Power: Graphs, Charts,
Schedules, and Maps
Pictographs,
pp. 20-31
Circle Graphs, pp. 32-43
Bar Graphs, pp. 44-55
Line Graphs, pp.56-67
Schedules and Charts, pp. 8495
Top 50 Math Skills for GED
Success Understanding Line of
Best Fit, pp. 76-77
Ex. Use an informal line of best fit to
interpret data, make predictions and draw
conclusions
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
121
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Use Measures of Central Tendency
and Other Statistical Measures
Ex. Identify, find and/or use the shape,
range, median, mean, and mode of data
Ex. Describe the effect of changes in the
data set on the mean and median and
discuss which is more representative of
the data
*Create*
In Lessons 7 and 8, students
will learn to:
 Develop strategies for
finding the median and
mean
 Use mean and median
to describe a data set
Achieving TABE Success D
Finding Mean, Median, and
Mode, p. 158
Math Sense: Measurement and Data
Analysis
Finding the Mean, p. 100
Finding the Median and Mode,
pp. 102-103
Choosing a Central Tendency,
p. 104
Gridding in Mean, Medians,
and Mode, pp. 112-113
Top 50 Math Skills for GED
Success
Apply Measures of Central
Tendency, pp. 70-71
Find a Missing Term when the
Mean is Known, pp. 72-73
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
122
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Use Probability
Ex. Recall and use basic probability
concepts to find the probability of
independent events happening
Ex. Find the probability of a single
outcome in a concrete situation and state
it as a ratio, fraction and percent
p.158 – Simple probability
(fractions)
p. 167 Complementary
Probabilities (fractions)
Get It Together
Spinners (Probability on
Spinners)
Which Spinner #1, p.104
Which Spinner #2, p. 105
Draw the Spinner #3, p. 109
Spinner #4, p. 110
Achieving TABE Success D
Investigating Probability, pp.
151-152
Ex. Find combinations and permutations
Top 50 Math Skills for GED
Success Probability, pp. 90-93
Review and Test Practice
Achieving TABE Success D
Probability, Data, and Statistics
Skills Checkup, pp. 164-165
Post-Assessment
Number Power: Graphs, Charts,
Schedule, and Maps
Graph Review, pp. 68-75
Schedule and Chart Review, pp.
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
123
Data, Statistics, and Probability
GED
EQUIPPED FOR THE FUTURE
STANDARD
EMPower
and
Unless stated otherwise, the
lessons below are from Many
Points Make a Point: Data and
Graphs (MPMP). Many of the
lessons in MPMP are beneficial
for GED students
CONTENT TOPICS

OACE Teacher Developed
Resources
The Math Problem Solver
This resource integrates number
development with data
interpretation. The particular
types of numbers involved are
noted in parentheses.
Electronic Resources
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Other Resources OACE
Teachers Currently Use
96-101
Suggested Resources for Developing Number Sense and Operation Proficiency at the BE1/2 Levels
To be sufficiently prepared to complete the BE1/2 Levels and begin the BE 3 Level, manage the mathematical demands of work and family, students should develop proficiency with the
following content topics related to Number Sense and Operations.
Equipped for the Future Standard:
Read, write and interpret a wide variety of mathematical information and concepts and apply to real-life and theoretical problems
EMPower
CONTENT TOPICS

Unless stated otherwise, the lessons come
from EMPower’s Everyday Number Sense:
Mental Math and Visual Models (ENS). These
lessons are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for BE
Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
WHOLE NUMBERS
WHOLE NUMBERS
Pre-Assessment
Use the Initial Assessment in the ENS
Teacher Book (pp. 169-173) to get a
sense of students’ general understanding
of operations and their skills in mental
math
Achieving TABE Success E
Skills Inventory Pretest: Part B Applied:
Computation in Contexts – Problems 10,
22, and 29
OACE AIM Data, Statistics, and Probability. Version 1 Fall 2011
124
Number Sense and Operations
BE 1/2
EMPower
CONTENT TOPICS

Applications and Problem Solving
Unless stated otherwise, the lessons come
from EMPower’s Everyday Number Sense:
Mental Math and Visual Models (ENS). These
lessons are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for BE
Levels 1/2.
Electronic Resources
Throughout the ENS book.
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
Get It Together
Hundred Chart
Tim’s Number, p. 18
Meg’s Number, p. 19
Paul’s Number, p. 20
Keisha’s Number, p. 21
Ex. Solve problems involving dollars and
cents
Ex. Use estimation to solve problems
and check reasonableness
Achieving TABE Success E
Adding Money, p. 33
Subtracting Money, p. 43
Math Sense: Whole Numbers and Money
Adding Dollars and Cents, pp. 66-67
Estimating Costs, pp. 68-69
Subtracting Dollars and Cents, pp. 94-95
Deciding to Add or Subtract, pp. 82-83
Figuring Change, pp. 100-101
Basic Notation, Rules of Order and
Properties
Ex. Write numbers in words (e.g.,
another way to write 5200 is five
thousand two hundred)
Ex. Use place value to create equivalent
representations of numbers through
three digits (such as 45 is the same as 4
tens and 5 ones or 20 + 20 + 5)
Ex. Identify the place value of digits in
the context of money (e.g., what is the
value of the 2 in $2565.00?)
In Lessons 1 & 2, students:
 Use mental math strategies to
estimate totals
 Mentally calculate by rounding
and adjusting
 Use mathematics notation to
describe mental math strategies
In Lessons 3-5, students:
 Use the number line as a
thinking tool
 Locate numbers on the number
line and determine the distance
Multiplication Resource The
Achieving TABE Success E
National Library of Virtual
Reading and Writing Numbers, p. 10
Manipulatives
Comparing Numbers, p. 11
http://nlvm.usu.edu/en/nav/frames
_asid_192_g_1_t_1.html
Using Patterns (Multiplication
facts)
TIAN Bundle Activity 1D, pp. 16–18
http://adultnumeracy.terc.edu/pdfs/
B1_Math_Topic.pdf
OACE AIM Number Sense and Operations. Version 1 Fall 2011
125
Number Sense and Operations
BE 1/2
EMPower
CONTENT TOPICS

Ex. Determine the relative size of
numbers of up to four digits
Ex. Recall or derive single-digit
multiplication facts
Ex. Use commutative property to create
equivalent representations of single-digit
multiplication facts (such as 5 x 3 is the
same as 3 x 5)
Ex. Use the inverse relationship between
addition and subtraction to add and
subtract whole numbers up to four digits
in a variety of problems including those
related to geometry, measurement, and
data
Ex. Choose appropriate strategies to
check for reasonableness of answers
Unless stated otherwise, the lessons come
from EMPower’s Everyday Number Sense:
Mental Math and Visual Models (ENS). These
lessons are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for BE
Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
(difference) between the
numbers. In Lesson 5, the
number line is extended to
include negative numbers
 Record mental math and
number line actions with
equations
In Lessons 6 & 7, students:
 Examine and identify the
composition of numbers in
terms of 10s, 100s, and 1000s
 Use mathematical notation,
including parentheses, to show
how they solve a problem
 Identify and use patterns when
multiplying or dividing by 10,
100, and 1000
In Lessons 8 & 9, students:
 Represent expressions using
arrays and/or groups arranged
to correspond with numbers
and operations
 Identify and find equivalent
expressions
 Record problem-solving
strategies using equations and
pictures
RATIONAL NUMBERS
Use Opening the Unit and the Initial
OACE AIM Number Sense and Operations. Version 1 Fall 2011
126
Number Sense and Operations
BE 1/2
EMPower
CONTENT TOPICS

RATIONAL NUMBERS
Pre-Assessment
Ex. Recognize that 50% is the same as ½
is the same as .5, and 25% is the same as
¼ is the same as .25
Ex. Recognize and use multiple
representations (pizza models, bars, tiles,
etc.) of one-half and one-quarter
Ex. Determine one-half and one-quarter
(and their equivalent forms 50% and 25%)
of an amount by halving and estimate
when amounts are close to these
benchmark fractions and percents
Ex. Add and subtract numbers involving
benchmark fractions
Unless stated otherwise, the lessons come
from EMPower’s Everyday Number Sense:
Mental Math and Visual Models (ENS). These
lessons are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for BE
Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
Assessment (pp. 87-91) in the Using
Benchmarks: Fractions, Decimals and
Percents Teacher book
In Lesson 1, students consider the
fraction ½ and:
 Identify the part and whole in
various cases
 Consider whether a fractional
amount is more than, less
than, or equal to ½.
 State the fraction that
represents the whole for any
case
No Matter What Shape Your
Fractions Are In
http://math.rice.edu/~lanius/Patter
ns
In Lesson 2, students:
 Develop methods for
calculating ¼ of a quantity
 Determine the amount “left
over” when ¼ is removed
In Lesson 3, the fraction ¾ is the
focus, and students:
 Develop methods for
calculating ¾ of a quantity
 Connect division and
multiplication with finding ¾
of a quantity
In Lesson 4, the three benchmarks
OACE AIM Number Sense and Operations. Version 1 Fall 2011
127
Number Sense and Operations
BE 1/2
EMPower
CONTENT TOPICS

PROPORTION
Ex. Recognize that a ratio remains the
same in simple proportions using
concrete representations (e.g. tiles,
pictures, diagrams, etc.)
Unless stated otherwise, the lessons come
from EMPower’s Everyday Number Sense:
Mental Math and Visual Models (ENS). These
lessons are aimed at Levels 3 and 4 students,
but can be adjusted in many cases for BE
Levels 1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
serve as referents between which
other fractions exist. Students:
 Compare fractions involving
numbers up to 1,000 to
determine where they are
located in relationship to the
benchmark (½, ¼, or ¾).
Keeping Things in Proportion: Reasoning
with Ratios, Lesson 1: A Close Look
at Supermarket Ads
Review and Test Practice
Post-Assessment
OACE AIM Number Sense and Operations. Version 1 Fall 2011
128
Number Sense and Operations
BE 3/4
Suggested Resources for Developing Number Sense and Operation Proficiency at the BE3/4 Levels
Whole Numbers and Integers
To be sufficiently prepared to complete the BE3/4 Levels and begin the GED level, manage the mathematical demands of work and family, and to anticipate preparation for post-secondary
readiness, students should develop proficiency with the following content topics related to Number Sense and Operations.
Equipped for the Future Standard:
Read, write and interpret a wide variety of mathematical information and concepts and apply to real-life and theoretical problems
EMPower
CONTENT TOPICS

WHOLE NUMBERS
WHOLE NUMBERS
Pre-Assessment
Unless stated otherwise, the lessons
come from EMPower’s Everyday Number
Sense: Mental Math and Visual Models
(ENS). These lessons are aimed at
Levels 3 and 4 students, but can be
adjusted in many cases for BE Levels
1/2.
Use Opening the Unit and the Initial
Assessment, pp. 169-173 in the ENS
Teacher Book, to get a sense of
students’ general understanding of
operations and their skills in mental
math
Throughout the ENS book.
Applications and Problem Solving
Ex. Add subtract, multiply, and divide
whole numbers to solve a variety of
problems, including those related to
money, geometry, measurement, and
data
Ex. Estimate solutions to problems
involving addition, subtraction,
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
Get It Together
Hundred Chart
Tim’s Number, p. 18
Meg’s Number, p. 19
Paul’s Number, p. 20
Keisha’s Number, p. 21
The Million Dollar Activity
TIAN Bundle #1, Number Sense:
Flexibility and Fluency, pp. 11–15
http://adultnumeracy.terc.edu/TIA
N_bundle1.html
About How Many?
TIAN Bundle #1, Number Sense:
Flexibility and Fluency, pp. 6–10
http://adultnumeracy.terc.edu/TIA
Achieving TABE Success M
Addition, pp. 15-23
Addition Skills Checkup, pp. 24-25
Subtractions, pp. 26-33
Subtraction Skills Checkup, pp. 3435
Multiplication, pp. 36-45
Multiplication Skills Checkup, pp.
46-47
Division, p. 48-60
OACE AIM Number Sense and Operations. Version 1 Fall 2011
129
Number Sense and Operations
BE 3/4
EMPower
CONTENT TOPICS

Unless stated otherwise, the lessons
come from EMPower’s Everyday Number
Sense: Mental Math and Visual Models
(ENS). These lessons are aimed at
Levels 3 and 4 students, but can be
adjusted in many cases for BE Levels
1/2.
N_bundle1.html
multiplication or division to determine
reasonableness of results
Basic Notation, Rules of Order and
Properties
Ex. Write numbers, up to six digits, in
words
Ex. Identify the place value of digits in
numbers up to millions
Ex. Use properties of numbers (magnitude
and order, place value, factors, multiples,
etc.)
Ex. Use the rules of order for all the
operations
Electronic Resources
In Lessons 1 & 2, students:
 Use mental math strategies to
estimate totals
 Mentally calculate by rounding
and adjusting
 Use mathematics notation to
describe mental math strategies
In Lessons 3-5, students:
 Use the number line as a
thinking tool
 Locate numbers on the number
line and determine the distance
(difference) between the
numbers. In Lesson 5, the
number line is extended to
include negative numbers
 Record mental math and
number line actions with
equations
In Lessons 6 & 7, students:
 Examine and identify the
composition of numbers in
The Four Operations: What Do
They Mean?
TIAN Bundle Activity 2A
http://adultnumeracy.terc.edu/pdfs
/B2_Math_Topic.pdf
Using Patterns (Multiplication
facts)
TIAN Bundle Activity 1D, pp. 16–18
http://adultnumeracy.terc.edu/pdfs
/B1_Math_Topic.pdf
Interactive Broken Calculator
Activity
http://www.mathsisfun.com/games
/broken-calculator.html
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
Division Skills Checkup, pp. 61-62
Achieving TABE Success M
The Number System
Reviewing Place Value, p. 8
Writing Whole Numbers in
Expanded Form, p. 9
Reading and Writing Whole
Numbers, p. 10
Comparing Whole Numbers, p. 11
Rounding Whole Numbers, p. 12
OACE AIM Number Sense and Operations. Version 1 Fall 2011
130
Number Sense and Operations
BE 3/4
EMPower
CONTENT TOPICS

Basic Notation, Rules of Order and
Properties (continued)
Unless stated otherwise, the lessons
come from EMPower’s Everyday Number
Sense: Mental Math and Visual Models
(ENS). These lessons are aimed at
Levels 3 and 4 students, but can be
adjusted in many cases for BE Levels
1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
terms of 10s, 100s, and 1000s
 Use mathematical notation,
including parentheses, to show
how they solve a problem
 Identify and use patterns when
multiplying or dividing by 10,
100, and 1000
In Lessons 8 & 9, students:
 Represent expressions using
arrays and/or groups
arranged to correspond with
numbers and operations
 Identify and find equivalent
expressions
 Record problem-solving
strategies using equations and
pictures
In Lessons 10-12, students:
 Focus on two models of
division and making sense of
remainders
 Solve division problems
involving splitting an amount
into equal parts
 Work with direct
measurement and scale to
find the number of groups of
a given size in a total
 Establish the relation
OACE AIM Number Sense and Operations. Version 1 Fall 2011
131
Number Sense and Operations
BE 3/4
EMPower
CONTENT TOPICS

Basic Notation, Rules of Order and
Properties (continued)
INTEGERS
Ex. Recognize and apply negative
integers in real contexts (such as
thermometers, winning/losing money, sea
level, etc.)
Unless stated otherwise, the lessons
come from EMPower’s Everyday Number
Sense: Mental Math and Visual Models
(ENS). These lessons are aimed at
Levels 3 and 4 students, but can be
adjusted in many cases for BE Levels
1/2.
Electronic Resources
OACE Teacher Developed
Resources
Other Resources OACE Teachers
Currently Use
For sharing information about lessons
and activities teachers have developed
and found to be effective
between division and
multiplication
 Identify factors of numbers
such as 48 and 72
 Find remainders and express
them using fractions,
decimals, and whole numbers
Lesson 5
The number line is extended to
include negative numbers
Ex. Recognize and represent negative
integers on a number line
Review and Test Practice
Post-Assessment
Closing the Unit and Final
Assessment (pp. 177-182), ENS
Teacher Book
OACE AIM Number Sense and Operations. Version 1 Fall 2011
132
Number Sense and Operations
BE 3/4
Suggested Resources for Developing Number Sense and Operation Proficiency at the BE3/4 Levels
Rational Numbers (Fractions, Decimals, and Percents) and Proportional Reasoning
To be sufficiently prepared to complete the BE3/4 Levels and begin the GED level, manage the mathematical demands of work and family, and to anticipate preparation for post-secondary
readiness, students should develop proficiency with the following content topics related to Number Sense and Operations.
Equipped for the Future Standard:
CONTENT TOPICS
Read, write and interpret a wide variety of mathematical information and concepts and apply to real-life and theoretical problems
EMPower
Key to…
See notes under “General Description”
series
Four EMPower Books address this
large area of study.
Using Benchmarks: Fractions, Decimals
and Percents and Split It Up: More
Fractions, Decimals and Percents lay an
important foundation for visualizing
and connecting benchmark fractions,
decimals, and percents. Operation Sense:
Even More Fractions, Decimals and
Percents develops conceptual
understanding of the four operations
with fractions and decimals, and
Keeping Things in Proportion: Reasoning
With Ratios develops proportional
reasoning.
Key To Fractions, Key to
Decimals, and Key to Percents
booklets primarily offer
skill practice
OACE Teacher Developed
Resources
Electronic Resources

RATIONAL NUMBERS and
PROPORTIONAL
REASONING
General Description
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
RATIONAL NUMBERS
Pre-Assessment
Start with the Initial Assessments in
Using Benchmarks and Split It Up to
OACE AIM Number Sense and Operations. Version 1 Fall 2011
Achieving TABE Success M
Decimal Skills Checkup,
133
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

determine a starting point for the
class.
Ex. Use benchmark fractions, decimals,
and percents (such as 3/4 and 1/10) to
estimate relative sizes
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
pp. 94-95
Using Benchmarks: Fractions, Decimals
and Percents
In Lesson 1, students consider the
fraction ½ and:
 Identify the part and whole in
various cases
 Consider whether a fractional
amount is more than, less than, or
equal to ½.
 State the fraction that represents
the whole for any case
The following provide
visual practice with the
benchmark fractions,
decimals, and percents
(Use in conjunction with
EMPower’s Using
Benchmarks and Split It Up)
In Lesson 2, students:
 Develop methods for calculating
¼ of a quantity
 Determine the amount “left over”
when ¼ is removed
Key to Decimals
Book 1: Decimal
Concepts
Key to Fractions
Book 1: Fraction
Concepts
Key to Percents
Book 1: Percent Concepts
In Lesson 3, the fraction ¾ is the
focus, and students:
 Develop methods for calculating
¾ of a quantity
 Connect division and
multiplication with finding ¾ of a
quantity
In Lesson 4, the three benchmarks
serve as referents between which
OACE AIM Number Sense and Operations. Version 1 Fall 2011
134
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
other fractions exist. Students:
 Compare fractions involving
numbers up to 1,000 to determine
where they are located in
relationship to the benchmark (½,
¼, or ¾).
In Lessons 5 & 6, the decimal 0.1 (.1)
is introduced. Students:
 Find one-tenth of a quantity
 Identify multiple ways of
representing one-tenth (1/10, 0.1,
and 10%) and relate them to
visual models
Split It Up: More Fractions, Decimals and
Ex. Extend benchmark fractions to
Percents
equivalent decimals and percents (1/10,
In Lesson 1, students:
1/100, etc.)
 Share ways to determine 10%, or
1/10, of an amount
Ex. Extend bank of benchmark fractions,
decimals, and percents (1/8, 1/6, etc.)
 Determine a total, given 10% of it
and understand how these relate on a
 Determine whether a variety of
number line
part-whole situations are more
than, less than, or equal to 10%
In Lesson 2, students:
 Share strategies to determine
multiples of 10% of a given
amount
 Base decisions involving percents
on the fact that the whole of an
amount or space equals 100%
 Show percents with arrays of 50
OACE AIM Number Sense and Operations. Version 1 Fall 2011
135
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
and 100 squares
 Name multiples of 10% with
equivalent fractions
In Lesson 3, students:
 Find 1% and its multiples of
three- and four-digit numbers
 Compare 10% of one amount and
1% of another to articulate the
effect of size of the percent and
the whole on the answer
In Lesson 4, students:
 Use multiples of 1% to find
single-digit percents
 Use multiples of 10% and 1%
combined to find two-digit
percents
In Lesson 5, students:
 Determine one-eighth of a given
amount
 Relate eighths to corresponding
percents
 Calculate 12 ½% (12.5%) using
multiples of 10% and 1%
 Demonstrate that 1/8 equals
125/1,000
In Lesson 6, students:
 Demonstrate thirds and their
OACE AIM Number Sense and Operations. Version 1 Fall 2011
136
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
percent equivalents using circle
graphs, number lines, and
countable objects
 Compare thirds to other
benchmark fractions
 Find the whole, given one-third or
two-thirds
In Lesson 7, students:
 Order corresponding fractions,
decimals, and percents
 Compare fractional amounts
 Choose a fraction, decimal, or
percent to solve problems
 Find whole amounts, given a part
Ex. Add, subtract and multiply numbers
involving benchmark fractions, decimals
and percents
In Lesson 8, students:
 Compare a percent with a
benchmark fraction
 Determine percents represented
by a part and a whole
 Find the combined percent of two
or more quantities
Operation Sense: Even More Fractions,
Decimals, and Percents
In Lesson 1, students:
 Reason about the equivalence of
fractions, decimals, and percents.
 Compare and order fractions,
decimals, and percents.
 Know that a/b can be interpreted
Key to Fractions
Book 3: Addition and
Subtraction
Cutting Cucumbers
TIAN Bundle #2 Activity
2B, pp. 7–11
http://adultnumeracy.ter
c.edu/pdfs/B2_Math_To
pic.pdf
Key to Decimals
What’s My Number?
http://mason.gmu.edu/~
Addition and Subtraction
Practice with Fractions and
Decimals:
OACE AIM Number Sense and Operations. Version 1 Fall 2011
137
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

as either a fraction (part/whole)
or as a division problem (the
quotient of a ÷ b).
In Lesson 2, students:
 Judge the reasonableness of
answers to addition problems
involving fractions, decimals, and
percents.
 Connect addition of fractions,
decimals, or percents to
combining quantities.
 Connect a picture or situation
with the math symbols.
 Pay attention to place value in
addition of fractions and decimals.
In Lesson 3, students:
 Interpret a subtraction problem in
three ways: as a take-away
situation, as a distance between
numbers, and as an absolute
comparison of two amounts.
 Use a number line to demonstrate
the difference between numbers
and absolute comparison.
In Lesson 4, students:
 Demonstrate understanding of
multiplication using pictures and
stories.
 Connect whole number
Book 2: Addition,
Subtraction, and
Multiplication.
mmankus/PBlocks/pbact
/whatnum.htm
--------------------------------
More Fun Fractions,
http://math.rice.edu/~la
nius/Patterns/add.html
Multiplication and
Division Practice with
Fraction and Decimals:
Key to Fractions
Book 2: Multiplying and
Dividing
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Drawing Fun Fractions
http://math.rice.edu/~la
nius/Patterns/draw.html
Key to Fractions
Book 4: Mixed Numbers
Key to Decimals
Book 3: Dividing
Key to Decimals
Book 4: Using Decimals
Key to Percents
Book 2: Percents and
Fractions
Key to Percents
Book 3: Percents and
Decimals
OACE AIM Number Sense and Operations. Version 1 Fall 2011
138
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources





Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
multiplication as repeated addition
to fraction and decimal
multiplication.
Distinguish among multiplication
by 1, by numbers less than 1, and
by numbers greater than 1.
Recognize that multiplication is
commutative for all numbers.
Understand that multiplying by
1/a is equivalent to dividing by a.
Understand that multiplying a
number by its reciprocal results in
1.
In Lesson 5, students:
 Extend understanding of the
model for integer division as an
act of splitting or dealing out an
amount to include fraction and
decimal amounts.
 Match the verbal language and
symbolic written notation for
division as splitting to a concrete
model.
 Compare a/b with b/a.
In Lesson 6, students:
 Extend understanding of the
quotitive model of division to the
domain of fractions and decimals.
 Use mathematical symbols and
diagrams to express and visualize
OACE AIM Number Sense and Operations. Version 1 Fall 2011
139
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
the action of division.
 Relate division to multiplication.
Achieving TABE Success M
Identifying Decimals, p.
84
Reading and Writing
Decimals, p. 85
Relating Decimals and
Fractions, p. 86
Recognizing Equivalent
Decimals, p. 87
Comparing Decimal
Numbers, p. 88
Adding Decimals, p. 89
Subtracting Decimals, p.
90
Multiplying Decimals, p.
91
Rounding Decimals, p. 92
Using Decimals to Solve
Word Problems, p. 93
Ex. Identify and use decimal place value
Ex. Create equivalent representations of
numbers up to billion and to the nearest
thousandth
Ex. Add and subtract decimals up to
three places
Pre-Assessment
Start with Opening the Unit and the
Assessments are on the
Initial Assessment in Keeping Things in last pages of each booklet
Proportion: Reasoning with Ratios Teacher
Book
Keeping Things in Proportion: Reasoning
With Ratios
In Lessons 1-3, students:
 Compare and construct equal
ratios, diagram the relationships,
write rules for creating equal
Cool Kool-aid
Experiment
http://alex.state.al.us/les
son_view.php?id=9860
OACE AIM Number Sense and Operations. Version 1 Fall 2011
Achieving TABE Success M
Writing Ratios, p. 78
Recognizing a Proportion,
p. 79
Writing a Proportion, p.
80
140
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

ratios, and estimate and predict
using ratios involved in shopping,
cooking, and figuring the time it
takes to do tasks.
Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
Solving a Proportion, p.
81
In Lesson 4, students
 Examine the difference between
ratios representing part:part and
those representing part:whole in
surveys. They focus on the
various ways to express ratios.
In Lessons 5-6, students
 Work with relationships involved
in resizing and redesigning twodimensional objects (such as
replicas of the Mona Lisa painting
and their personal calculators).
They add graphing as a tool for
examining the proportions.
In Lesson 7-9, students
 Explore the new tools of unit rate
and the cross-product property in
order to compare and create ratios
with more difficult numbers. They
work with speed and currency
conversions.
In Lesson 10, students
 Use percents as a convenient way
OACE AIM Number Sense and Operations. Version 1 Fall 2011
141
Number Sense and Operations
BE 3/4
CONTENT TOPICS
EMPower
Key to…
See notes under “General Description”
series
OACE Teacher Developed
Resources
Electronic Resources

Other Resources OACE
Teachers Currently Use
For sharing information about
lessons and activities teachers
have developed and found to be
effective
to compare the relationships
within and between data sets of
different sizes. national and local
demographics of educational
programs provide the context.
Review and Test Practice
Post-Assessment
Closing the Unit and Final
Assessment (pp. 159-163) in the
Keeping Things in Proportion Teacher
Book
OACE AIM Number Sense and Operations. Version 1 Fall 2011
142
Number Sense and Operations
GED
Suggested Resources for Developing Number Sense and Operation Proficiency at the GED Level
To be sufficiently prepared to pass the GED, manage the mathematical demands of work and family, and to begin preparation for post-secondary readiness, students should develop
proficiency with the following content topics related to Number Sense and Operations.
Equipped for the Future Standard:
Read, write and interpret a wide variety of mathematical information and concepts and apply to real-life and theoretical problems
CONTENT TOPICS
EMPower

Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
WHOLE NUMBERS
WHOLE NUMBERS
Pre-Assessment
Use the Final
Assessment in Everyday
Number Sense: Mental
Math and Visual Models
Teacher Book (pp. 191200) to get a sense of
students’ general
understanding of
operations and their
skills in mental math.
Assign lessons
accordingly – see
BE3/4.
pp. 23-24
pp. 34-35
pp. 36-37
pp. 75-76
The Four Operations:
What Do They Mean?
TIAN Bundle Activity 2A
http://adultnumeracy.te
rc.edu/pdfs/B2_Math_
Topic.pdf
Multiplication
Resource
The National Library of
Virtual Manipulatives
http://nlvm.usu.edu/en
/nav/frames_asid_192_
g_1_t_1.html
Interactive Broken
Calculator Activity
http://www.mathsisfun.
OACE AIM Number Sense and Operations. Version 1 Fall 2011
143
Number Sense and Operations
GED
CONTENT TOPICS
EMPower

Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
com/games/brokencalculator.html
Applications and Problem Solving
Ex. Add subtract, multiply, and divide
whole numbers to solve a variety of
problems.
Test practice at the end
of each lesson in Keeping
Things in Proportion
Lessons 1-4 give
students in GED-level
prep programs the
opportunity to use
addition and subtraction
of whole numbers in a
variety of problem
solving situations that
include geometry and
algebra.
Lessons 7-10 focus on
multiplication and
division of whole
numbers in a variety of
problem solving
situations that include
geometry and algebra.
Lesson 11 covers
Powers and Roots
Achieving TABE Success
D
Problem Solving
Following a five-step
plan, p. 32
Identifying the Question
to Answer, p. 33
Identifying too Much or
too Little Information,
p. 34
Choosing an Operation
and Solving a Problem,
p. 35
Solving Two-Step Word
Problems, p. 36
Using Estimation to
Solve Word Problems,
p. 37
Problem Solving Skills
Checkup, pp. 38-39
Number Power:
Word Problems
OACE AIM Number Sense and Operations. Version 1 Fall 2011
144
Number Sense and Operations
GED
CONTENT TOPICS
EMPower

Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
Addition and
Subtraction Word
Problems: Whole
Numbers, pp. 18-27
More Addition and
Subtraction Word
Problems: Whole
Numbers, pp. 28-39
Multiplication and
Division Word
Problems: Whole
Numbers, pp. 61-76
Applications and Problem Solving
(continued)
Basic Notation, Rules of Order and
Properties
Ex. Use place value and/or the
commutative, associative, and
distributive properties to create
equivalent representations of whole
numbers
Ex. Determine squares and square
roots to solve problems such as those
related to geometry
Ex. Draw on prior knowledge with
powers of 10 to understand scientific
notation
INTEGERS
INTEGERS
pp. 56-57
OACE AIM Number Sense and Operations. Version 1 Fall 2011
145
Number Sense and Operations
GED
CONTENT TOPICS
EMPower
Key To Algebra

The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
Pre-Assessment
Book 1: Operations on
Integers
Ex. Add subtract, multiply, and divide
integers to solve a variety of
problems
Ex. Understand how and why the
rules of addition and subtraction work
with negative integers
Ex. Draw on knowledge of subtraction
and the number line to understand
absolute value
RATIONAL NUMBERS
RATIONAL NUMBERS
Pre-Assessment
Ex. Use benchmark fractions,
decimals and percents to make
estimates and to check for
reasonableness
Use the Final
Assessment in the
Operation Sense Teacher
Book (pp. 107-110) to
get a sense of students’
general understanding of
operations and their
skills in mental math
with the common
fractions. Assign lessons
accordingly – see BE
3/4
Lessons 15-17 focus on
fractions
Lesson 18 connects
fractions and decimals
Cutting Cucumbers
TIAN Bundle #2 Activity
2B, pp. 7–11
http://adultnumeracy.te
rc.edu/pdfs/B2_Math_
OACE AIM Number Sense and Operations. Version 1 Fall 2011
Number Power:
Word Problems
Word Problem Pretest,
pp. 1-7
146
Number Sense and Operations
GED
CONTENT TOPICS
EMPower

Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
Topic.pdf
Ex. Read, write, and compare
fractions and mixed numbers and
decimals
What’s My Number?
http://mason.gmu.edu/
~mmankus/PBlocks/p
bact/whatnum.htm
Ex. Add, subtract, multiply and divide
fractions to solve a variety of
problems
More Fun Fractions
http://math.rice.edu/~l
anius/Patterns/add.html
Ex. Add, subtract, multiply and divide
decimals to solve a variety of
problems
Drawing Fun
Fractions
http://math.rice.edu/~l
anius/Patterns/draw.ht
ml
Ex. Multiply and divide decimals and
determine to what degree the result is
valid when used with measurements
Ex. Reason as to how elements of
operations with fractions are reflected
in the same operation with decimals
The Target Game
Connected Mathematics
Project
http://connectedmath.
msu.edu/CD/Grade6/
Benchmark/index.html
#rules
PROPORTION
PROPORTION
Pre-Assessment
Use the Final
Assessment in the
Keeping Things in
Proportion Teacher Book
(pp. 159-163) to get a
OACE AIM Number Sense and Operations. Version 1 Fall 2011
147
Number Sense and Operations
GED
CONTENT TOPICS
EMPower

Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
sense of students’
mastery of proportion.
Lessons 20-21 focus on
ratio and proportion
Ex. Build on understanding of ratios,
to include equivalent forms of
benchmark fractions (such as 2/4 =
1/2)
Ex. Find ratios
Ex. Use proportional reasoning to
solve a variety of problems, including
percent increase or decrease
Suggest GED students
explore at least 3-4
investigations in Keeping
Things in Proportion, such
as:
Lesson 2: It’s a Lot of
Work!
Lesson 5: Mona Lisa, Is
That You?
Lesson 7: Comparing
Walks
Lesson 9: The Asian
Tsunami
Cool Kool-aid
Experiment
http://alex.state.al.us/les
son_view.php?id=9860
Larger Than Life
TIAN Bundle #5,
Geometric Thinking, pp. 9–
11
http://adultnumeracy.ter
c.edu/TIAN_bundle5.ht
ml
Comparing Measures
with the Proportioner
http://seeingmath.conco
rd.org/interactive_docs/
PR_Activity.pdf
OACE AIM Number Sense and Operations. Version 1 Fall 2011
Achieving TABE Success
D
Writing Ratios, p. 81
Finding Equal Rations,
p. 82
Achieving TABE Success D
Using Ratio and Percent
to Solve Word Problems,
p. 96
Number Power:
Word Problems
Using Proportions to
Solve Word Problems,
pp. 93-95
Using Proportions to
Solve Decimal Word
Problems, pp. 96-98
Using Proportions to
Solve Faction Word
Problems, pp. 98-99
Solving Conversion Word
Problems, pp. 100-101
148
Number Sense and Operations
GED
CONTENT TOPICS
EMPower

Ex. Solve a variety of problems
involving percents
Key To Algebra
The Math Problem
Solver
OACE Teacher
Developed Resources
Electronic Resources
For sharing information
Other Resources OACE about lessons and activities
Teachers Currently Use teachers have developed
and found to be effective
Lesson 23 focuses on
Percents
Achieving TABE Success D
Solving Three Types of
Percent Problems, pp.
94-95
Number Power:
Word Problems
Percent Word Problems
Identifying the Parts of a
Percent Word Problem,
pp. 118-120
Solving Percent Word
Problems, pp. 121-126
Percent and Estimation,
pp. 127-128
The Percent Circle, pp.
129-131
Solving Percent Word
Problems with Decimals
and Fractions, pp. 132134
Solving Percent Word
Problems, pp. 135-136
Review and Test Practice
Post-Assessment
OACE AIM Number Sense and Operations. Version 1 Fall 2011
149
List of Resources
Title
Publisher
Publication
Date
ISBN
EMPower
McGraw-Hill
2012
See detailed list at
Level(s)
BE 1/2, 3/4,
GED
end of chart
Key to …
Key Curriculum
1980
Download complete
BE 3/4, GED
order form from: http://
Press
www.keypress.com/Doc
uments/KeyToorderFor
m.pdf
Achieving TABE
Contemporary
2006
978-0-07-704467-1
BE 1/2
Contemporary
2006
978-0-07-704468-8
All (BE 1/2, use
success in
Mathematics E
Achieving TABE
success in
selectively)
Mathematics M
Achieving TABE
Contemporary
2006
978-0-07-704469-5
success in
GED, BE 3/4
(use selectively)
Mathematics D
GED Exercise
Contemporary
2002
978-0-8092-2237-7
GED
Get It Together:
Lawrence Hall of
1989
0-912511-53-2
All
Math Problems
Science
2003
1-56420-381-6
BE 3/4
2003
1-56420-122-5
All
Books:
Mathematics
for Groups
Math Sense:
New Readers
Algebra and
Press
Geometry
Math Sense:
New Readers
Measurement
Press
150
Title
Publisher
Publication
Date
ISBN
Level(s)
BE 1/2, 3/4,
GED
Math Sense:
New Readers
2003
1-56420-383-2
All
Whole Numbers
Press
2003
1-56420-385-94
All
Contemporary
1988
978-0-07-659229-6
BE 3/4, GED
Contemporary
1988
978-0-07-659233-3
BE 3/4, GED
Contemporary
1988
978-0-07-659231-9
BE 3/4, GED
Contemporary
1988
978-0-07-659230-2
BE 3/4, GED
Contemporary
1988
978-0-07-659236-4
BE 3/4, GED`
Steck-Vaughn
2009
978-1-419-05354-
BE 3/4, BE 1/2
22
(use selectively)
and Data Analysis
and Money
Math Sense:
New Readers
Comprehensive
Press
Review
Number Power:
Geometry
Number Power:
Analyzing Data
Number Power:
Word Problems
Number Power:
Graphs, Charts,
Schedules, and
Maps
Number Power:
Review
TABE
Fundamentals
LEVEL M
Applied Math
TABE
Steck-Vaughn
2009
978-1-419-05355-9
BE 3/4
Steck-Vaughn
2009
978-1-419-05358-0
GED, BE 3/4
Fundamentals
LEVEL M
Math
Computation
TABE
151
Title
Publisher
Publication
Date
ISBN
Fundamentals
Level(s)
BE 1/2, 3/4,
GED
(use selectively)
LEVEL D
Applied Math
TABE
Steck-Vaughn
2009
978-1-419-05359-7
Fundamentals
GED, BE 3/4
(use selectively)
LEVEL D
Math
Computation
Top 50 Math
McGraw-Hill
2004
978-0071445221
GED
McGraw-Hill
2003
0072943009
GED
New Readers
2005
978-156420-460-8
BE 3/4, GED
2004
978-156420-459-2
BE 3/4, GED
2005
978-156420-461-5
BE 3/4, GED
Skills for GED
Success
The Math
Problem Solver
Visual Literacy -
Scales, Charts and Press
Diagrams
Visual Literacy –
New Readers
Tables and
Press
Graphs
Visual Literacy –
New Readers
Maps,
Press
Photographs, and
Editorial Cartoons
152
EMPower series - Teacher Books
Title
Many Points Make a Point: Data and
Graphs
Seeking Patterns, Building Rules:
Algebraic Thinking
Over, Around, and Within: Geometry
and Measurement
Using Benchmarks: Fractions,
Decimals, and Percents
Split It Up: More Fractions, Decimals,
and Percents
Everyday Number Sense: Mental Math
and Visual Functions
Keeping Things in Proportion:
Reasoning with Ratios
Operation Sense: Even More
Fractions, Decimals, and Percents
ISBN
978-0-07662-095-1
978-0-07662-096-8
978-0-07662-097-5
978-0-07662-098-2
978-0-07662-099-9
978-0-07662-100-2
978-0-07662-101-9
978-0-07662-102-6
EMPower series - Student Books
Title
Many Points Make a Point: Data and
Graphs
Seeking Patterns, Building Rules:
Algebraic Thinking
Over, Around, and Within: Geometry
and Measurement
Using Benchmarks: Fractions,
ISBN
978-0-07662-087-6
978-0-07662-088-3
978-0-07662-089-0
978-0-07662-090-6
153
Decimals, and Percents
Split It Up: More Fractions, Decimals,
and Percents
Everyday Number Sense: Mental Math
and Visual Functions
Keeping Things in Proportion:
9780--07662-091-3
978-0-07662-092-0
978-0-07662-093-7
Reasoning with Ratios
Operation Sense: Even More
Fractions, Decimals, and Percents
EMPower series – FULL SET
978-0-07662-094-4
978-0-07662-488-1
Includes 1 copy of each Teacher book, and 10 copies of each student book
OACE Teacher Developed Resources
GED Practice
Steve
Test
Meyerson
2002
BE 4/5
2004
BE 4/5
2007
BE 4/5
2007
BE 4/5
questions
PA, PB, PC
GED Math
Steve
Practice
Meyerson
questions II
PE, PD
GED Practice
Steve
Test
Meyerson
questions
PF and PG
Practice
Steve
Questions for
Meyerson
154
TABE MATH
D9 and D10
GED Math Kit
and PRE-GED
Math Kit
155
The Four Big Ideas
Communication
“Instructional programs from pre-kindergarten through grade 12 should enable all students to –
 organize and consolidate their mathematical thinking through communication;
 communicate their mathematical thinking coherently and clearly to peers, teachers, and
others;
 analyze and evaluate the mathematical thinking and strategies of others;
 use the language of mathematics to express mathematical ideas precisely” (Principles and
Standards for School Mathematics, 2000, p. 60).
“Use math to solve problems and communicate” (Equipped for the Future Standards Wheel,
2001).
Connections
“Instructional programs from pre-kindergarten through grade 12 should enable all students to –
 recognize and use connections among mathematical ideas;
 understand how mathematical ideas interconnect and build on one another to produce a
coherent whole;
 recognize and apply mathematics in contexts outside of mathematics” (Principles and
Standards for School Mathematics, 2000, p. 64).
“A high quality mathematics curriculum for adult learners should feature worthwhile tasks, such
as activities that are drawn from the context of real life experience” (Adult Numeracy Network’s
Teaching and Learning Principles, 2005).
Richer Definition of Mathematics Proficiency
“Recognizing that no term captures completely all aspects of expertise, competence, knowledge,
and facility in mathematics, we have chosen mathematical proficiency to capture what we
believe is necessary for anyone to learn mathematics successfully. Mathematical proficiency, as
we see it, has five components, or strands:
conceptual understanding—comprehension of mathematical concepts, operations, and
relations
procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately
strategic competence—ability to formulate, represent, and solve mathematical problems
adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
productive disposition—habitual inclination to see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence and one’s own efficacy” ( National
Research Council, Adding It Up: Helping Children Learn Mathematics, 2001, p, 115)
156
“A high quality mathematics curriculum for adult learners should weave together all the
elements of mathematical proficiency – not only procedural fluency, but also conceptual
understanding, ongoing sense-making, problem solving, and a positive attitude about learning
mathematics” (Adult Numeracy Network’s Teaching and Learning Principles, 2005).
All Strands at All Levels
The National Council of Teachers of Mathematics emphasizes that the Content Standards of
Number, Algebra, Geometry, Measurement, and Data Analysis and Probability apply across all
grade levels, even though they should receive different emphases across the grade bands”
(Principles and Standards for School Mathematics, 2000, p. 30).
Number
Algebra
Geometry
Measurement
Data Analysis and
Probability
“A high quality mathematics curriculum for adult learners should include the concepts of number,
data, geometry, and algebra at all levels of learning so that students can develop and connect
mathematical ideas” (Adult Numeracy Network’s Teaching and Learning Principles, 2005).
157
Teacher’s Plans for and Reflections on Lesson, using the Four Big Ideas. Please write in the learning objectives
before teaching the lesson and complete the rest of the instrument after teaching the lesson.
Name:
Date:
Lesson:
Materials:
What do you want students to learn and be able to do? (Write down learning objectives in the spaces below before teaching the lesson. After
teaching the lesson, rate the degree to which you feel each learning objective was addressed.)
Rating (0 – 3)
Learning Objectives
0 – none at all
1 – a bit
2 – adequate
3 – emphasized strongly
Students will learn/be able to:



Did students learn or demonstrate an ability to do something I did not anticipate or plan to emphasize? Were my original learning objectives
appropriately challenging and/or attainable for the range of levels of students in my class?
158
Assess your lesson based on the five strands of mathematical proficiency.
Rating (0 – 3)
Strand
Conceptual understanding
Meaning and Guiding Questions
Comprehension of mathematical concepts, relations, and operations
To what extent did I:
Encourage students to use multiple representations and understand how different
representations relate to each other?
Help students understand the meaning behind the math symbols and their use?
Help students connect the mathematical ideas being studied to what they previously
studied and what they already know?
Connect the math ideas to familiar life situations?
Comments:
Procedural fluency
(Procedures refer to “step-bystep” methods for accomplishing
a mathematical task, such as
long division, subtracting mixed
numbers, converting from
decimals to percents, computing
the area of a triangle, etc.)
Comments:
Strand
Strategic competence
Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
To what extent did I:
Clearly explain and demonstrate the procedure?
Help students understand when they should use a particular procedure?
Encourage the use of estimation to check for the reasonableness of an answer?
Have students consider the efficiency of a particular procedure?
Help students understand the mathematics behind a procedure?
Meaning and Guiding Questions
Rating (0 – 3)
Ability to formulate, represent, and solve problems
To what extent did I:
Give students practice solving a wide variety of problems?
Teach strategies explicitly?
Talk with students about how to use different strategies for different situations?
Comments:
159
Adaptive reasoning
Capacity for logical thought, reflection, explanation, and justification
To what extent did I:
Ask students to explain how they solved a problem?
Ask students to consider how others’ explanations were similar to or different from
their own?
Encourage students to use a sequence of logical steps to arrive at a conjecture, an
answer, or a conclusion?
Use, or encourage my students to use, analogies to make connections?
Encourage my students to use different ways to justify an answer?
Comments:
Productive disposition
Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy
To what extent did I
Encourage and equip students to tackle problems that were challenging for them?
Take steps to establish a learning environment in which students feel that their
ideas are respected and valued?
Make the lesson engaging for my students?
Manage the classroom environment to improve learning for all?
Comments:
Use a scale of 0 – 3 to rate the degree to which “Making Connections” was addressed.
Meaning and Guiding Questions
Rating (0 – 3)
To what extent did I
Help students connect the mathematical ideas to goals and life roles?
Make connections to other mathematics ideas?
Make connections to other subject areas?
Help students make connections to familiar life experiences?
Comments:
160
Use a scale of 0 – 3 to rate the degree to which “Communication” was addressed.
Meaning and Guiding Questions
Rating (0 – 3)
To what extent did I
Have students work together to investigate and solve problems?
Ask open-ended questions?
Encourage students to ask questions and to share their thinking?
Provide opportunities for students to explain their work to each other and to
the whole class?
Comments:
Use a scale of 0 – 3 to rate the degree to which “All Strands at All Levels” was addressed.
Meaning and Guiding Questions
Rating (0 – 3)
To what extent did I:
Integrate math content across two or more content strands?
Adapt the lesson to different levels?
Have students investigate math ideas intuitively, informally and/or formally?
Find accessible ways to expose students to math content that they typically
would not be taught until much later?
Comments:
Further Thoughts to Consider.
Knowing what I know now, what, if anything, would I do differently?
What is my next step with this class?
161
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