This PowerPoint is from day 4 of
Math Week. It covers…
1. Differentiation
2. The math of Unit 5
3. The math of part 4 of Unit 4
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High School Math
The Standards Based Way
Day 4
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Nicole Spiller
West Georgia RESA
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Problem of the Day
2 + 2, then f(g(2)) =
If
f(
x)
=
and
g(x)
=
2x
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A) 3 B) 5
C) 7
D) 7 5/9
E) 16 2/3
The graphs of y = f(x) and y = g(x) for 0 to 10
are shown in the figure below. For how many
values of x is the product f (x) g( x) = 0 for 0 to 10?
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A) Two B) Four C) Five D) Six E) Seven
Housekeeping
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•
•
•
•
Breaks
Cell Phones
Restrooms
Parking Lot
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Essential Question/Enduring
Understandings
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• What is the Math of Unit Five?
• What is the Math of Part 4 of Unit Four?
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• How can I use differentiation in a HS SBC?
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• How can the Math Support Class support
differentiation?
Activator
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• Quick Talk
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Differentiation is…
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• Focused on essential understandings, principles,
concepts, and skills
• Designed to provide respectful, meaningful, and
engaging work
• Flexible
• Qualitative, not Quantitative
• Student centered
• Dynamic
• Rooted in Assessment
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A Working Definition
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• Differentiation is:
– A teaching philosophy based on the premise
that teachers should adapt instruction to student
differences
– Students have multiple options for acquiring
content, processing and making sense of ideas
and developing products so each student can
learn successfully.
– Handout – Comparing Classrooms
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Preparing for Differentiation
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• Begin Small – Do not feel rushed to differentiate
every minute off every day
• Understand one’s self – Teachers are better able to
differentiate if they understand themselves as
learners
• Start with favorite and familiar topics – It is easier
if you are already comfortable with the content
• Consider working in teams – Other teachers give
support and encouragement and add knowledge as
well as an experience base
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- Applied Differentiation: Making it work on the classroom.
School Improvement Network. 2005.
Managing Differentiation
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• Sustain a positive climate – A welcoming, safe, positive
climate is essential because differentiating instruction
requires that students take risks as well as their teachers
• Provide anchor activities – These activities engage students
when they finish assigned work or do not know what to do
• Establish and practice routines – Routines help students
know when to give their attention to the teacher and how to
begin and end activities.
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Applied Differentiation: Making it work on the classroom. School
Improvement Network. 2005.
Tiered Lessons
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• A tiered lesson is a differentiation strategy
that addresses a particular standard, key
concept, and generalization, but allows
several pathways for students to arrive at an
understanding of these components, based
on the students’ interests, readiness, or
learning profiles.
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Cheryll Adams, Ph.D. Ball State University
Partner Reading
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• Team up with a partner
• Partner A – Read “What is equity and how is
it evident in mathematics classrooms?”
• Partner B – Read “How can different
learning styles be address with consistent
expectations?”
• Share your thoughts from your reading with
your partner, then we will discuss as a group.
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We have a choice…
We can teach to the middle
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and hope for the best.
OR
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We can accommodate the full diversity of
academic needs and accept the challenge of
diversity by taking student readiness, student
learning/personality profiles, and student
interests into consideration.
-Unknown
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Paula’s Peaches
A Launching Task
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• This task has, as a major emphasis, the
ability to solve simple non-linear equation
as well as the concepts of: domain, range,
zeros, intercepts, intervals of increase and
decrease, maximums and minimums, end
behavior, as well as the idea that an
equation can be seen as two functions set
equal to each other where the ‘answers’ are
the intersection points.
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Paula’s Peaches
A Launching Task
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• Factoring Quadratics with a leading
coefficient of 1
• May need to review, functions have not
been emphasized since Unit Two
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Part 4 of Unit Four
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• Card, Marble, Dice, and Simulation Tasks
• Transition from independent to dependent events
• Use of histograms and frequency tables to
calculate probabilities
• Create and compare experimental to theoretical
probabilities
• Perform simulations on the Graphing Calculator
• Transition to large sample spaces
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Unit 5 – Algebra in Context
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• Prerequisites
– Extensive work with operations on integers,
rational numbers, an square roots of nonnegative numbers
– Concepts of similarity and transformations
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Overview of Unit 5
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• An application and extension all of the
algebra standards listed as key standards
addressed in Units 1 and 2
• Solving equations via factoring
• Viewing solutions as points of intersection
of graphs
• Solving basic radical equations
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Overview of Unit 5
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• Techniques for solving rational equations
with a denominator that is a rational
number or a 1st degree polynomial
• Connections of graphs and solutions
• An emphasis on the concept of finding
equivalent equations, exceptions to this
concept, and the concept of solution sets
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Essential Question/Enduring
Understandings
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• There is an important distinction between solving an
equation and solving an applied problem modeled by an
equation. The situation that gave rise to the equation may
include restrictions on the solution to the applied problem
that eliminate certain solutions to the equation.
• The definitions of even and odd symmetry for functions
are stated as algebraic conditions on values of functions,
but each symmetry has a geometric interpretation related to
reflection of the graph through one or more of the
coordinate axes.
• For any graph, rotational symmetry of 180 degrees about
the origin is the same as point symmetry of reflection
through the origin.
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Essential Question/Enduring
Understandings
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• Techniques for solving rational equations include steps that
may introduce extraneous solutions that do not solve the
original rational equation and, hence, require an extra step
of eliminating extraneous solutions.
• Understand that any equation in can be interpreted as a
statement that the values of two functions are equal, and
interpret the solutions of the equation domain values for
the points of intersection of the graphs of the two
functions. In particular, solutions of equations of the form
f(x) = 0, where f(x) is an algebraic expression in the
variable x, correspond to the x-intercepts of the graph of
the equation y = f(x).
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Key Standards
MM1A1. Students will explore and interpret the
characteristics of functions, using graphs, tables, and
simple algebraic techniques.
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c. Graph transformations of basic functions including vertical shifts,
stretches, and shrinks, as well as reflections across the x- and yaxes.
d. Investigate and explain characteristics of a function: domain,
range, zeros, intercepts, intervals of increase and decrease,
maximum and minimum values, and end behavior
h. Determine graphically and algebraically whether a function has
symmetry and whether it is even, odd or neither.
i. Understand that any equation in x can be interpreted as the
equation f(x) = g(x), and interpret the solutions of the equation
as the x-value(s) of the intersection point(s) of the graphs of y =
f(x) and y = g(x).
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Key Standards
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MM1A2. Students will simplify and operate with radical
expressions, polynomials, and rational expressions.
a. Simplify algebraic and numeric expressions involving
square root.
b. Perform operations with square roots.
c. Add, subtract, multiply, and divide polynomials.
d. Add, subtract, multiply, and divide rational expressions.
e. Factor expressions by greatest common factor, grouping,
trial and error, and special products limited to the formulas
listed.
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Key Standards
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MM1A3. Students will solve simple equations.
Solve quadratic equations in the form ax2 + bx + c = 0 where
a = 1, by using factorization and finding square roots where
applicable.
Solve equations involving radicals such as, using algebraic
techniques.
Use a variety of techniques, including technology, tables,
and graphs to solve equations resulting from the
investigation of .
Solve simple rational equations that result in linear equations
or quadratic equations with leading coefficient of 1.
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Related Standards
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• MM1P1: Students will solve problems (using appropriate
technology)
• MM1P2: Students will reason and evaluate mathematical
arguments
• MM1P3: Students will communicate mathematically
• MM1P4: Students will make connections among
mathematical ideas and to other disciplines
• MM1P5: Students will represent mathematics in multiple
ways
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Concepts/Skills to Maintain
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• Operations on integers, rational numbers,
and square roots of non-negative numbers
• Basic properties of basic quadratic, cubic,
absolute value, square root, and rational
functions
• Adding, subtracting, multiplying, and
dividing elementary polynomial, rational, and
radical expressions
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Tasks
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• Task 1: A launching task that formalizes
factoring and properties of graphs
• Tasks 2-4: Techniques for solving rational
equations where the denominator is limited to
rational numbers and first degree polynomials.
Graph symmetry, odd and even functions,
solving radical and rational equations.
• Task 5: Culminating task that used applications
of geometry, distance formula and topics
learned in this lesson.
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Logo Symmetry
A Learning Task
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• Key Points:
–
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Review of line and rotational symmetry
Introduction of point symmetry
Introduction of even and odd functions
Reflection through the x and y axis
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Resistance
A Learning Task
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• Key Points
– Finding a more complex denominator
– Elimination of solutions due to physical
constraints (domain restrictions)
– Extraneous solutions
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Shadows and Shapes
A Learning Task
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• Key Points:
– Use idea of similar triangles
– Apply Pythagorean theorem
– Solutions to radical equations
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Fairfield Aviation
A Culminating Task
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• This task incorporates all of the material
learned in this unit in an applied setting.
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• It may be appropriate for students to work
on this task throughout the unit with
periodic deadlines
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End of Day 4
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