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Math and The Mind’s Eye
Common Core State Standards Correlation Charts
Unit 13 Sketching Solutions to Algebraic Equations
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Unit 13 Sketching Solutions to Algebraic Equations
A-SSE.1a 1. Interpret expressions that represent a quantity in
terms of its context. a. Interpret parts of an expression, such as
terms, factors, and Coefficients.
Activity 5
Irrational Roots
Activity 4
Equations Involving Rational
Expressions
Activity 3
Sketching Quadratics Part II
Activity 2
Sketching Quadratics Part 1
Standard
Activity 1
Sketching Solutions
The focus of this unit is to develop students ability to sketch a model of a mathematical situation. In this instance the situations involve proportions, classic Algebra problems, as
well as quadratics.. Providing students with a method for creating a visual model to solve complicated problems is at the heart of the Common Core State Standards.
Comments
This unit provides the classroom teacher with a detailed method
for teaching students how to sketch, and model, mathematical
situations, which is precisely what early mathematicians did when
investigating mathematical phenomena.
A-SSE.1b b. Interpret complicated expressions by viewing one or
more of their parts as a single entity. For example, interpret P(1+r)
n as the product of P and a factor not depending on P.
A-SSE.2 Use the structure of an expression to identify ways
to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
Activity 5 extends the modeling of equations with rational
solutions into modeling of irrationals.
A-SSE.3a Choose and produce an equivalent form of an
expression to reveal and explain properties of the quantity
represented by the expression. a. Factor a quadratic expression to
reveal the zeros of the function it defines.
Activities 2 and 3 provide opportunities for students to sketch
rectangles for quadratic expressions, where the area is the
expression and the dimensions are the factors.
A-SSE.3b b. Complete the square in a quadratic expression to
reveal the maximum or minimum value of the function it defines.
Completing the square can be modeled by determining what size
manipulative piece would fit in a model of an expression that is
"almost" a square.
A-APR.1 Understand that polynomials form a system analogous
to the integers, namely, they are closed under the operations
of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
A-CED.1 Create equations and inequalities in one variable and use
them to solve problems. Include equations arising from linear and
quadratic functions, and simple rational and exponential functions.
A-CED.4 Rearrange formulas to highlight a quantity of interest,
using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V IR to highlight resistance R.
A-REI.3 Solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
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Activity 5
Irrational Roots
Activity 4
Equations Involving Rational
Expressions
Activity 3
Sketching Quadratics Part II
Activity 2
Sketching Quadratics Part 1
Standard
Activity 1
Sketching Solutions
Unit 13 Sketching Solutions to Algebraic Equations continued
Comments
A-REI.4a Solve quadratic equations in one variable. a. Use the
method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula from this form.
A-REI.4b b. Solve quadratic equations by inspection (e.g., for x2
= 49), taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex
solutions and write them as a ± bi for real numbers a and b.
A-REI.5 Prove that, given a system of two equations in two
variables, replacing one equation by the sum of that equation and
a multiple of the other produces a system with the same solutions.
A-REI.6 Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.
A-APR.6 Rewrite simple rational expressions in different forms;
write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of
b(x), using inspection, long division, or, for the more complicated
examples, a computer algebra system.
A-APR.7 (+) Understand that rational expressions form a system
analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.
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