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18 | eureka Math alGeBra I Study GuIde
In Geometry, this practice is explicitly demonstrated in the way trigonometry is
introduced and developed in Module 2. In Lesson 25, students complete an Exploratory
Challenge that helps them recognize how the ratios of the length of the opposite side to
the hypotenuse and the length of the adjacent side to the hypotenuse are the same for
corresponding angles in similar triangles, which leads to the formalizing of trigonometric
ratios. In addition, the theme of approximation in Module 3 is an interpretation of
structure. Students approximate both area and volume for polyhedral regions. They must
understand how and why it is possible to create upper and lower approximations of a
figure’s area or volume. The derivation of the volume formulas for cylinders, cones, and
spheres and the use of Cavalieri’s principle are also based entirely on understanding the
structure and substructures of these figures.
In Precalculus, students recognize that the structure of matrices can help them identify
the geometric transformations induced by them; for instance, they can identify matrices
that induce pure dilations or pure rotations. They also recognize how to use the
structure of Pascal’s triangle to complete binomial expansions.mp.8. Look for and express regularity in repeated reasoning.
One of the main themes of this mathematical practice is that mathematics is open to
drawing conjectures from completing several related exercises, looking for regularity in
both what you have done and in the results you obtain.
In Algebra I, students solve equations in one variable in several different formats. From
the patterns they recognize, students look for general methods for solving a generic
linear equation in one variable by replacing the numbers with letters: ax+=bcxd+. They
can apply the method developed to rearrange formulas to isolate a variable of interest.
This practice is demonstrated in Module 4 of Geometry, specifically when students derive
a formula used to calculate the midpoint of a line segment based on repeated reasoning
from numeric examples.
In Algebra I and Algebra II, students recognize regularity in factoring patterns and
develop identities to express that regularity, including identities for the difference
of squares and cubes, the sum of squares, and the square of a sum and square of a
difference. In Module 2 of Algebra II and Module 4 of Precalculus, students form
conjectures about properties of trigonometric functions and develop formulas from the
regularities they recognize, including angle sum and difference formulas for sine and
cosine and the double angle formulas.
In summary, the Instructional Shifts and the Standards for Mathematical Practice help
establish the mechanism for thoughtful sequencing and emphasis on key topics in A Story of
Functions. It is evident that these pillars of the new standards combine to support Eureka
Math with a structural foundation for the content. Consequently, A Story of Functions is
artfully crafted to engage teachers and students alike while providing a powerful avenue for
teaching and learning mathematics.