Using positive and negative numbers in context mathematical goals

1. Using Positive and Negative Numbers in Context
Mathematical goals
This lesson unit is intended to help you assess how well
students are able to understand and use directed numbers in
context. It is intended to help identify and aid students who
have difficulties in ordering, comparing, adding, and
subtracting positive and negative integers. Particular attention
is paid to the use of negative numbers on number lines to
explore the structures:
Starting temperature + Change in temperature = Final
temperature Final temperature – Change in temperature =
Starting temperature Final temperature – Starting temperature =
Change in temperature
Introduction
The lesson unit is structured in the following way:
· Before the lesson students work individually on an assessment
task designed to reveal their current understanding. You review
their responses and write questions to help them improve their
work.
· After a whole-class introduction, students work together in
small groups on a card-matching task.
· In the same small groups, they then create directed number
calculations to correspond to temperature changes or city
temperatures. To end the lesson there is a whole-class
discussion.
· In a follow-up lesson, students again work alone on a task
similar to the assessment task.
Materials required
· Each student will need a copy of the assessment
task Temperature Changes and Temperature Changes
(revisited), a copy of the Temperature Scale, a mini-whiteboard,
a pen, and an eraser.
· Each small group of students will need a set of cut-up cards
from the sheets City Temperatures and Changes in Temperature,

2. a glue stick, and a large sheet of paper for making a poster.
· You will also need one copy of the cut-up cards Introduction
to Negative Numbers.
Time needed
15 minutes before the lesson, a 100-minute lesson (or two 55-
minute lessons), and 20 minutes in a follow-up lesson. Timings
are approximate and will depend on the needs of your class.
Lesson Type
· CConcept Development
Mathematical Practices
This lesson involves a range of mathematical practices from the
standards, with emphasis on:
· MP1:Make sense of problems and persevere in solving them
· MP2:Reason abstractly and quantitatively
· MP3:Construct viable arguments and critique the reasoning of
others
· MP4:Model with mathematics
· MP6:Attend to precision
· MP7:Look for and make use of structure
· MP8:Look for and express regularity in repeated reasoning
Mathematical Content Standards
This lesson asks students to select and apply mathematical
content from across the grades, including the content standards:
· 7.EE:Use properties of operations to generate equivalent
expressions.
· 7.EE:Solve real-life and mathematical problems using
numerical and algebraic expressions and equations.
· 7.NS:Apply and extend previous understandi ngs of operations
with fractions to add, subtract, multiply, and divide rational
numbers.
Compose Your Program and Learning Objectives Scoring Guide
CRITERIA
NON-PERFORMANCE
BASIC
PROFICIENT
DISTINGUISHED

3. Choose the appropriate level of Bloom’s Taxonomy verb for
each learning objective or goal.
26%
Does not choose a Bloom’s Taxonomy verb for each learning
objective or goal.
Chooses a verb that does not quite reflect the desired level of
Bloom’s Taxonomy verb for one or more learning objectives or
goals.
Chooses the appropriate level of Bloom’s Taxonomy verb for
each learning objective or goal.
Chooses the appropriate level of Bloom’s Taxonomy verb for
each learning objective or goal. Justifies the choice with sound
reasoning.
Compose three discrete program objectives or goals, relating
each to the intent of the learning experience to be designed.
27%
Does not compose program objectives or goals.
Composes less than three discrete program objectives or goals;
or does not relate each goal to the intent of the learning
experience to be designed; or does not adequately describe the
relationship between goals and course intent.
Composes three discrete program objectives or goals, relating
each to the intent of the learning experience to be designed.
Composes three discrete program objectives or goals, relating
each to the intent of the learning experience to be designed in a
clear and concise manner.
Compose one to three learning objectives or goals aligned to
each of the three program objectives or goals, relating each to
the intent of the learning experience to be designed.
27%
Does not compose learning objectives or goals.
Composes learning objectives or goals but does not align them
appropriately to the program objectives or goals or does not
relate each to the intent of the learning experience to be
designed.
Composes one to three learning objectives or goals aligned to

4. each of the three program objectives or goals, relating each to
the intent of the learning experience to be designed.
Composes one to three learning objectives or goals aligned to
each of the three program objectives or goals, relating each to
the intent of the learning experience to be designed in a clear
and concise manner.
Convey clear meaning through appropriate word choice and
usage.
10%
Does not convey clear meaning through appropriate word choice
and usage.
Clear meaning is inhibited by a limited vocabulary, vague or
ambiguous word choice, or incorrect usage.
Convey clear meaning through appropriate word choice and
usage.
Conveys precise and unequivocal meaning through precise word
choice and effective usage.
Conveys purpose, in an appropriate tone and style,
incorporating supporting evidence and adhering to
organizational, professional, and scholarly writing standards.
10%
Does not convey purpose, in an appropriate tone and style,
incorporating supporting evidence and adhering to
organizational, professional, and writing scholarly standards.
Conveys purpose, in an appropriate tone or style. Clear,
effective communication is inhibited by insufficient supporting
evidence and/or minimal adherence to applicable writing
standards.
Addresses assignment purpose in a well-organized text,
incorporating appropriate evidence and tone in grammatically
sound sentences.
Conveys clear purpose, in a tone and style well-suited to the
intended audience. Supports assertions, arguments, and
conclusions with relevant, credible, and convincing evidence.
Exhibits strict and nearly flawless adherence to organizational,
professional, and scholarly writing standards, including APA

5. style and formatting.
Week 3 Best Practices
One of the four conditions essential to motivation, whether in
site-based learning environments or the online environment, is
maximizing inclusion among adult learners. While there are
differences between the two environments, there are even more
similarities. This week you engage in both inductive and
deductive reasoning to analyze how those similarities and
differences are manipulated in a learning environment and how
to accommodate those needs in learning experiences.
Integrated Design
Fink, L. D. (2013). Creating significant learning experiences:
An integrated approach to designing college courses (Rev.
ed.). San Francisco, CA: Jossey-Bass:
· In Chapter 3, “Designing Significant Learning Experiences I:
Getting Started,” on pages 67–112, you will explore the
beginning steps in using the integrated design.
Encouraging Motivation
Wlodkowski, R. J., & Ginsberg, M. B. (2017). Enhancing adult
motivation to learn: A comprehensive guide for teaching all
adults (4th ed.). San Francisco, CA: Jossey-Bass:
· In Chapter 5, “Encouraging Motivation in Online Formats,” on
pages 107–146, you will learn about the needs of online learners
and specific strategies for facilitating motivation in the online
environment.
· In Chapter 6, “Establishing Inclusion Among Adult Learners,”
on pages 147–182 you will focus on specific strategies that
promote the core need of inclusion among adult learners,
including factors related to culture, neuroscience and onsite
versus online environments.

6. Professional Learning Module
Modeling the enactment of the Formative Assessment:
Using Positive and Negative Numbers in Context (Concept
Development)
This Professional Learning Module describes how a facilitator
can implement, with fidelity, the Shell Center’s
Formative Assessment Lesson, Using Positive and Negative
Numbers in Context, so that participants
understand how to enact, in their own classrooms, this lesson
that is compatible with the Common Core State
Standards (CCSS). Resources for each component of the module
are included as page numbers in the Lesson
Guide, and as links to additional professional learning tools
such as handouts and videos.
Shell Center’s Lesson Guide:
http://map.mathshell.org/materials/lessons.php?taskid=453&sub
page=concept
Framing the lesson
Teachers frame the lesson so that students know what to expect
throughout the entire process, from the pre- to
the post-lesson assessment. Participants learn what it means to
frame a lesson, why it is important, and what
might go wrong if it is neglected.
Handout: Framing a Formative Assessment Lesson
Temperature Changes: the pre-lesson assessment
The pre-lesson assessment is designed to surface the common
issues that impede student learning. We have
found it important to ask students to complete it individually in
class, without help from other students or their
teacher. Coaching students to the right answers can keep
common issues hidden and hinder student success.
The pre-lesson assessment is not graded, but is analyzed to
identify student errors. These data are used by

7. teachers to develop feedback questions or comments, and to
modify instruction.
Lesson Guide: page T-2; assessment S-1 & S-2; Facilitator
Notes: Administering the Pre-Lesson Assessment
Whole-class interactive introduction
For successful enactment of the introduction it is important to
follow the Lesson Guide’s suggestions exactly.
Replacing the whole-class discussion with a mini-lesson will
quickly sabotage the entire lesson.
Lesson Guide: pages T-4; Temperature Scales S-3, card set S-4;
slide Temperature Changes P-1
Introducing the collaborative activity
By taking care to introduce the collaborative activity teachers
give students the chance to wrap their arms
around the activity that they are being asked to do. Students
need to be given clear instructions in writing as
well as access to these written instructions for the duration of
the activity so that they can grapple with the
complexities of the activity and maximize their learning.
Lesson Guide: page T-5; slide Instructions for Working
Together P-2
Engaging in the collaborative activity
The collaborative activity is designed as an opportunity for
students to learn. This gives students an
opportunity to engage in many of the CCSS Standards for
Mathematical Practice—for example MP2 (Reason
abstractly and quantitatively) and MP3 (Construct viable
arguments and critique the reasoning of others). This
opportunity to learn is enhanced when teachers give students
feedback, allow students to struggle productively
in homogeneous pairs, and mini-conference so as to scaffold
student learning.
Lesson Guide: pages T-5, T-6 & T-7; card sets S-5, S-6 & S-7;

8. slides More Temperature Changes P-3 & P-4;
Videos: A Teacher Listens, The Importance of Homogeneous
Grouping & Mini-Conference
Temperature Changes (revisited): the post-lesson assessment &
sharing prepared feedback
Students are given back their pre-lesson assessments and their
teachers’ feedback questions and comments
before being asked to complete the post-lesson assessment. The
post-lesson assessment gives students the
chance to demonstrate growth across the pre- and post-lesson
assessments. We have found it important to
ask students to complete it individually in class, without help
from other students or their teacher.
Lesson Guide: page T-8; assessment S-8 & S-9; Facilitator
Notes: Administering the Post-Lesson Assessment
Whole-class discussion
This is the most difficult part of almost any lesson. Teachers
who have enacted the Shell Center’s Formative
Assessment Lessons tell us that this is the part of the lesson
where they feel the most vulnerable, and the part
where they find the Lesson Guide’s specific advice on what they
might say to their students most useful.
Lesson Guide: page T-7 & T-8; Video: Whole-class discussion
Ann Shannon & Associates, LLC
Professional Learning Module
Following up on the enactment of the Formative Assessment
Lesson:
Using Positive and Negative Numbers in Context (Concept
Development)

9. This Professional Learning Module describes how to follow up
on the enactment of the lesson Using Positive
and Negative Numbers in Context, and how to facilitate the
professional learning opportunities that are
afforded by Formative Assessment Lessons.
Shell Center’s Lesson Guide:
http://map.mathshell.org/materials/lessons.php?taskid=453&sub
page=concept
Resource: Facilitator Notes
The mathematics of Using Positive and Negative Numbers in
Context
Participants reflect on the lesson to determine its mathematics.
They consider its alignment to the CCSS and
reflect upon how the lesson presents students with an
opportunity to learn.
Lesson Guide: page T-1; Handout: The Mathematics of Using
Positive and Negative Numbers in Context
Analyze student responses to the pre-lesson assessment
Participants analyze a small set of student responses to the pre -
lesson assessment to identify and name the 3
to 5 most important common issues evident in the student work.
Participants record these data in a copy of the
Growth Analysis Spreadsheet.
Lesson Guide: pages T-3, & T-8; Student work: Temperature
Changes;
Spreadsheet: Growth Analysis Spreadsheet; Handout: How to
use the Student Analysis Growth Spreadsheet
Write feedback questions and comments that will move the
learner forward
Using the Criteria for Feedback handout, participants practice
developing 3 to 5 feedback questions that are
designed to encourage students to think more deeply about the
common issues and move their learning

10. forward. This written feedback is shared with students at the
beginning of the lesson to help them re-engage
with the problem.
Handout: Criteria for Feedback
Analyze student responses to the post-lesson assessment
Participants analyze a small set of student responses to the post-
lesson assessment and add these data to
their copy of the Growth Analysis Spreadsheet in order to
illustrate student growth across the pre- and post-
lesson assessments.
Lesson Guide: page T-9; Student work: Temperature Changes
(Again)
Modifying subsequent instruction
Assessment is not formative until it is used to modify
subsequent instruction. The formative assessment
lessons are designed to reveal evidence about student learning.
Here participants use all of the evidence
—the evidence revealed during the lesson and that collected
from the assessments—to modify subsequent
instruction so that it better promotes student learning and is
compatible with the CCSS.
Link the structure of the lessons to the theory of formative
assessment
The Big Idea and the Five Strategies studied earlier convey the
interpretation of formative assessment that
underpins the Shell Center’s lessons. Participants consider how
the structure of the lessons maps on to the theory.
Handout: Big Idea of Formative Assessment; Handout: Five
Strategies of Formative Assessment
When to enact this lesson in your classroom
Deciding when to enact a Formative Assessment Lesson can be
a challenge. Many lessons fit well about two-

11. thirds of the way through a compatible unit of instruction, but
would also work well as part of a review. The
Course Outline provides a suggestion for when to enact this
lesson.
Resource: A Course Outline for Grade 7
Ann Shannon & Associates, LLC