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16 | eureka Math alGeBra II Study GuIde
identities do or do not hold. In cases where they decide that the given student work is
incorrect, students work to develop the correct general algebraic results and justify them
by reflecting on what they perceived as incorrect about the original student solution.mp.4. model with mathematics.
One interpretation of this standard is to ensure that students see that mathematics is
useful. A Story of Functions contains modeling lessons throughout, where students apply
a modeling process to a real-world situation. The specific modeling lessons are indicated
in the Topic Overviews for each course.
Algebra I Module 5 develops students’ modeling skills and requires students to be able to
represent real-world situations with equations and graphs. A specific example involves
modeling the cooling of an object with an exponential decay equation (this situation is
revisited in Algebra II). In Module 4 of Geometry, students model the motion of a robot in
the plane to determine the extent of motion within the bounds of a room and to move the
robot to the location of the source of a beacon signal in the infinite plane. In Algebra II,
students apply exponential models to explore mortgages and home loans, as well as
investments that will help them pay off credit card debt and accrue savings over time. They
also use trigonometric equations to model the motion of a Ferris wheel. In Precalculus,
students use trigonometric functions to model the motion of waves. They use vectors to
model the compressive forces distributed along stone arches to determine the effect of
base column height and buttressing on the stability of the arch. They also apply their
understanding of geometric representations of linear transformations to explore how
three-dimensional objects are projected onto two-dimensional screens. In Module 4, they
apply their understanding of trigonometry to determine the best viewing height of objects.mp.5. use appropriate tools strategically.
Using appropriate tools strategically must be interpreted broadly to include many
options for students. In this curriculum, using appropriate tools strategically could
include the use of tape diagrams, tabular models, transparencies, protractors, logarithm
tables, random number generators, spreadsheet software, graphing applications,
standard normal tables, rulers, and compasses.
Building students’ independence with the use of models and tools is important, as is
empowering students to determine when it is appropriate to use a specific tool. In
Algebra I, students are introduced to the tabular model that can be applied to multiply
polynomials and in the reverse to factor them. It is up to students to decide when this
tool can be used efficiently and when it may be more efficient for them not to use
this tool. Students need to decide for what sample size it makes sense to calculate
measures of center and spread by hand and when it is more efficient to use a spreadsheet
or graphing calculator. They also use graphing calculators to generate residual plots,
which help them determine whether a function used to model a data set is appropriate.
In Geometry, students need to decide what tools to use to complete constructions and
geometric transformations—for instance, when they may need tracing paper to carry out
a rigid transformation or when it might be most effective to explore the properties
of objects using software. They need to determine when it is appropriate to calculate
the exact measures of objects and when they may need to use a calculator to find a
numerical estimate. In Algebra II, students may need to decide when it is beneficial to use
graph paper to create the graph of a function versus using graphing software, which may
enable them to more accurately produce graphs in three dimensions or graphs of