

The
fourphase cycle of classroom activities,
is designed to actively engage students in investigating and making sense of problem situations, in constructing important mathematical concepts and methods, in generalizing and proving mathematical relationships, and in communicating, both orally and in writing, their thinking and the results of their efforts. Most classroom
activities are designed to be completed by students working collaboratively
in groups of two to four students punctuated by wholeclass discussions
(that retain the cognitive demand of problems) once students
have had some time to discuss ideas and have partial understanding
of the mathematical ideas being developed. 
Short clips of classrooms are available. 
Think About This
Situation
The lesson launch promotes a teacherled discussion of a problem situation
and of related questions to think about. This discussion sets the context
for the student work to follow and helps to generate student interest;
it also provides an opportunity for the teacher to assess student knowledge,
to discuss cultural and mathematical context suppositions, and to clarify
directions for the investigation to follow.
Sample launch
Investigation
Classroom activity then shifts to investigating focused problems and
questions related to the launching situation by gathering data, looking
for patterns, constructing models and meanings, and making and verifying
conjectures. As students collaborate in pairs or small groups, the
teacher circulates among students providing guidance and support,
clarifying or asking questions, giving hints, providing encouragement,
and drawing group members into the discussion to help groups collaborate
more effectively. Periodically, it may be valuable to facilitate
wholeclass discussions, particularly at the close of the class period.
The investigations and related questions posed by students drive
the learning.
Sample exploration
Resource for developing collaboration skills—Dysfunctional Group Skit
Summarize the
Mathematics
This investigative work is followed by a teacherled class discussion
(referred to as Summarize the Mathematics) in which students summarize
and explain the reasoning supporting mathematical ideas developed in
their groups, providing an opportunity to construct a shared understanding
of important concepts, methods, and approaches. This discussion leads
to a class summary of important ideas or to further exploration of a
topic if competing perspectives remain. Varying points of view and differing
conclusions that can be justified should be encouraged. This discussion
based on student thinking is crucial to building understanding of mathematical
ideas for the procedural skill development to follow.
Sample Summarize the Mathematics
Check
Your Understanding
Students are given an individual task to complete and reinforce their
understanding of concepts and methods. Teachers should encourage students
to develop the ability to determine for themselves whether their understanding
is enough to secure progress to the tasks found in the On Your Own homework
tasks.
Sample Check Your Understanding
On
Your Own Homework Sets
In addition to the classroom investigations, CorePlus Mathematics provides
sets of On Your Own tasks, which are designed to engage students in applying,
connecting, reflecting on, extending, and reviewing their evolving mathematical
knowledge. On Your Own tasks are provided for each lesson in the materials
and are central to the learning goals of each lesson. These tasks are
intended primarily for individual work outside of class. Selection of
homework tasks should be based on student performance and the availability
of time and technology. Also, students should exercise some choice of
tasks to pursue, and, at times should be given the opportunity to pose
their own problems and questions to investigate. The chart below describes
the types of tasks in a typical On Your Own set.
On Your Own: Homework Tasks  
Applications  These tasks provide opportunities for students to use and strengthen their understanding of the ideas they have learned in the lesson. 
Connections  These tasks help students to build links between mathematical topics they have studied in the lesson and to connect those topics with other mathematics that they know. 
Reflections  These tasks provide opportunities for students to reexamine their thinking about ideas in the lesson. 
Extensions  These tasks provide opportunities for students to explore further or more deeply the mathematics they are learning. 
Review  These tasks provide opportunities for justintime review and distributed practice of mathematical skills to maintain procedural fluency. 
These tasks in the OYO are intended primarily for individual work outside of class. If a few students are identified as needing additional assistance with specific skills, they should be given additional assistance outside of class. Selection of homework tasks should be based on student performance, the availability of time, and technology access. Also, students should exercise some choice of tasks to pursue, and at times should be given the opportunity to pose their own problems and questions to investigate.
Practicing for
Standardized Tests
Opportunities for additional review and practice are provided in the
Practicing for Standardized Tests masters in each Unit Resources booklet.
Each Practicing for Standardized
Tests master presents 10 questions that draw on all content
strands. The questions are presented in the form of test items similar
to how they often appear in standardized tests such as state assessment
tests, the Preliminary Scholastic Aptitude Test (PSAT), SAT, ACT PLAN,
or ACT. We suggest using these practice sets following the unit assessment
so students can become familiar with the formats of standardized tests
and develop effective testtaking strategies for performing well on such
tests.
Preparing for
Undergraduate Mathematics Placement Tests
Most colleges and universities administer a mathematics placement test
to incoming students. The results of this test together with grades in
high school mathematics courses and intended undergraduate major are
used to advise students regarding the first college mathematics course
they study. About half of these postsecondary institutions allow students
the use of technology (e.g., a graphing calculator) while completing
the test. But you may wish to ensure that your students are prepared
to take placement tests with and without technology.
To perform well on these tests, there are
a number of skills and reasoning techniques that need to be automatic.
Developing that level of proficiency requires practice. Thus, you will
find two Preparing for
Undergraduate Mathematics Placement (PUMP) sets in the unit resources
for each of the Course 4 units.
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