14 | eureka Math alGeBra I Study GuIde
Type III: For these problems, students must model real-world situations using mathematics
and demonstrate more advanced problem-solving skills.
how A Story of functionS aLigns with the standards for
mathematiCaL praCtiCe
Like the Instructional Shifts, each Standard for Mathematical Practice is integrated into
the design of A Story of Functions.
mp.1. make sense of problems and persevere in solving them.
An explicit way in which the curriculum integrates this standard is through its
commitment to consistently engage students in solving multi-step problems. For
example, in Algebra I, students make sense of real-world problems involving linear and
exponential growth. They analyze the critical components of the problem, presented as
a verbal description, a data set, or a graph, and persevere in writing the appropriate
function that describes the relationship between two quantities. In Module 3, students
are sometimes required to write functions to represent data using a linear model and an
exponential model and to compare how well each function equation represents the data.
They select the most appropriate model and use it to make predictions. In Geometry,
students work on a recurring problem throughout Module 4 in describing the motion of a
robot bound within the room in which it sits. Through perseverance, students discover
the slope criteria for perpendicular and parallel lines, the means to find the coordinates
of a point dividing a line segment into two lengths in a given ratio, and the distance
formula of a point from a line. In Algebra II, students solve rational and radical equations,
which require them to consider the possibility of extraneous solutions. They also make
sense of quadratic equations that do not contain real number solutions, coming to
understand that the complex number system provides solutions to the equation x^2 += 10
and higher-degree equations. In Algebra II and Precalculus, students represent real-
world situations with systems of equations, solve the systems using algebraic means or
inverse matrix operations (in Precalculus), and examine the reasonableness of solutions.
Purposeful integration of a variety of problem types that range in complexity naturally
invites students to analyze givens, constraints, relationships, and goals. Problems require
students to organize their thinking, which necessitates critical self-reflection on the
actions they take to problem-solve. On a more foundational level, concept sequence,
activities, and lesson structure present information from a variety of novel perspectives.
The question, “How can I look at this differently?” undergirds the organization of the
curriculum, each of its components, and the design of problems.mp.2. reason abstractly and quantitatively.
An example of reasoning abstractly is seen in the analysis of graphs of functions in
Module 1 of Algebra I. Students analyze graphs of non-constant rate measurements and
infer from the shape of the graphs the quantities being displayed for a given situation and
suggest appropriate units for the quantities. Throughout the lessons in Algebra I Module
5, students are required to decontextualize information provided as data or in a verbal
description to analyze situations that can be represented using linear, quadratic, or