Fractions/Ratios/Rational
Expressions
High School Break-out Sessions
Quick diversion before we
start…
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decompressor
are needed to see this picture.
High School Break Out
Session Overview
Standards on Fractions/Ratios/Rational
Expressions
• What’s in the K-8 Standards?
– Progression based on Critical Areas
• A look 9-12. What and “where” are the
standards
• Discussion on specific standards /
sample problems and activities to get
students thinking deeply
K-8 Standards
• Standards Handout
• Critical areas in the Framework at the
elementary grades.
K-8 Standards
Grade 3
Critical Area #2
Students develop an understanding of fractions,
beginning with unit fractions, and they use fractions
along with visual fraction models to represent parts of
a whole. Students understand that the size of a
fractional part is relative to the size of the whole. For
example, 1/2 of the paint in a small bucket could be
less than 1/3 of the paint in a larger bucket, but 1/3 of
a ribbon is longer than 1/5 of the same ribbon because
when the ribbon is divided into 3 equal parts, the parts
are longer than when divided into 5 equal parts.
Students are able to use fractions to represent
numbers equal to, less than, and greater than one.
They solve problems that involve comparing fractions
by using visual models and strategies based on
noticing equal numerators and denominators.
K-8 Standards
Grade 4
Critical Area #2
Students develop understanding of fraction equivalence
and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and
they develop methods for generating and recognizing
equivalent fractions. Students extend previous
understandings about how fractions are built from unit
fractions, composing fractions from unit fractions,
decomposing fractions into unit fractions, and using the
meaning of fractions and the meaning of multiplication to
multiply a fraction by a whole number.
K-8 Standards
Grade 5
Critical Area #1
Students apply their understanding of fractions and
fraction models to represent the addition and subtraction
of fractions with unlike denominators as equivalent
calculations with like denominators. They develop
fluency in calculating sums and differences of fractions,
and make reasonable estimates of them. Students also
use the meaning of fractions, of multiplication and
division, and the relationship between multiplication and
division to understand and explain why the procedures
for multiplying and dividing fractions make sense. (Note:
this is limited to the case of dividing unit fractions by
whole numbers and whole numbers by unit fractions.)
K-8 Standards
Grade 6
Critical Area #2
Students use the meaning of fractions, the meanings
of multiplication and division, and the relationship
between multiplication and division to understand and
explain why the procedures for dividing fractions make
sense. Students use these operations to solve
problems. Students extend their previous
understandings of number and the ordering of
numbers to the full system of rational numbers, which
includes negative rational numbers, and in particular
negative integers. They reason about the order and
absolute value of rational numbers and about the
location of points in all four quadrants of the
coordinate plane.
K-8 Standards
Grade 6-8
Critical Areas address:
Grade 6
• Solve ratio and rate problems about quantities
• Connect ratios and fractions
• Solve a wide variety of problems involve ratios and
rates.
Grade 7
• Students extend ratios to proportions
• Solve proportion problems involve percent (discount,
interest, taxes, tips, etc.
• Scale drawing problems
• Unit rate as a measure of steepness, called slope
Grade 8…(no critical area - proportional relationships
graphically, decimal conversions, rational
approximations to irrationals)
Lets do Some Mathematics
• An Example of Rational Functions from
the Core-Plus Program Course 4
– Unit 3 - Algebraic Functions and Equations
Activity 1: Rational Function Models
9-12 Standards
• Read through the 9-12 Standards
specific to fractions/ratios/rational
expressions/fractions
– Any surprises?
– Any questions?
– Any concerns?
• Page 75 of the Frameworks -introduction to Number and Quantity
includes an overview of Number
development K-8.
Topping off student’s knowledge
of Rational Numbers
N-RN.3
• What
Explain why the sum or product of
rational numbers is rational; that the
sum of a rational number and an
irrational number is irrational; and that
the product of a nonzero rational
number and an irrational number is
irrational.
would an explanation for this look like?
• What would/should we accept/expect?
Extending Exponents to
Rational Exponents
N-RN.1
N-RN.2
Explain how the definition of the meaning of
rational exponents follows from extending the
properties of integer exponents to those
values, allowing for a notation for radicals in
terms of rational exponents. For example, we
define 5(1/3) to be the cube root of 5 because
we want (51/3)3 = 5(1/3x3) to hold, so (51/3)3
must equal 5.
Rewrite expressions involving radicals and
rational exponents using the properties of
exponents.
Extending Exponents to
Rational Exponents
• What’s the big deal about rational exponents
and how do we get students to see it?
• Why does the “rule” given below make
sense?
m
m
n
n
b

 b
• How much do we do without a calculator and
why?
Rational Expressions
A.APR.1a.MA Divide polynomials.
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.
q(x) + r(x)/b(x)
• Why is this form important? What’s the
point? What’s the goal? What does it
do for us?
• Try it!
– Rewrite the functions below in the form
above.
– Graph the functions.
– Do you see any benefits of this form?
4x  3
c(x) 
x 1
x  3x  1
t(x) 
x2
2
Rational Functions
F.IF.10.MA
Given algebraic, numeric, and/or graphical
representations of functions, identify the function as
polynomial, rational, logarithmic, exponential, or
trigonometric.
F.IF.7d (+)
Graph rational functions, identifying zeros and
asymptotes when suitable factorizations are
available, and showing end behavior.
F.IF.8
Write a function defined by an expression in different but
equivalent forms to reveal and explain different
properties of the function.
Write a function
Write the equation for a rational function…
1. that looks exactly like the line y = 2x+3 but with a
hole at x = 4.
2. with a horizontal asymptote of y = 2 and a vertical
asymptote of x = -3
3. with a horizontal asymptote of y = -3, a vertical
asymptote of x = 4 and a hole at x = 2.
4. that approaches the line y = 2x as |x| gets large, but
has a vertical asymptote at x = 0.
Write a function
10
5
5
-5
-10
-15
Write a function
10
Avoid the
brick walls
5
-10
Pass through
the “gates”
10
-5
-10
Problem adapted from
SIMMS-Integrated
Mathematics, unit
“Functioning on a Path”,
Level 4, Kendhall-Hunt,
2006
Write a Function
Problem taken from SIMMS-Integrated Mathematics, unit “Functioning on a Path”,
Level 4, page 110, Kendhall-Hunt, 2006
Rational Equations
A.REI.2 Solve simple rational and radical
equations in one variable, and give
examples showing how extraneous
solutions may arise.
Rational Functions
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.REI.11
Explain why the x-coordinates of the points where the
graphs of the equations y = f(x) and y = g(x) intersect are
the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions,
make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are linear, polynomials,
rational, absolute value, exponential, and logarithmic
functions.*
Rational Functions
F.BF.4
Find inverse functions.
F.BF.4a
Solve an equation of the form f(x) = c for a simple function f
that has an inverse and write an expression for the inverse.
For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x ≠ 1.
F.IF.10.M
A
Given algebraic, numeric, and/or graphical representations
of functions, identify the function as polynomial, rational,
logarithmic, exponential, or trigonometric.
Rational expressions/functions
Real World Examples?
• Take any area/volume formula and solve for
one of the dimensions to get examples of
inverse variation and inverse variation
squared.
• The area of a trapezoid is 20 square cm. The
length of one its bases is 3 cm. Write an
equation giving the height in terms of the area
of the other base.
Rational expressions/functions
- Modeling variation/combined variation
What is the maximum weight that can be held by a board
with dimensions t, d, and w as shown below?
Find a function that gives the maximum weight, M, in terms
of t, d and w.
Problem from Advanced Algebra, University of Chicago School
Mathematics Project, Scott Foresman Addison Wesley, 1998., p. 116-118.
Rational expressions/functions
- Modeling variation/combined variation
w (in)
1
2
3
4
5
6
M (lb)
27
53
80
107
133
160
t (in)
1
2
3
4
5
6
M (lb)
20
80
180
320
500
720
d (in)
1
2
3
4
5
6
M (lb)
800
400
267
200
160
133
Problem from Advanced Algebra, University of Chicago School
Mathematics Project, Scott Foresman Addison Wesley, 1998., p. 116-118.
Rational expressions/functions
- Modeling variation/combined variation
Problem from Advanced Algebra, University of Chicago School
Mathematics Project, Scott Foresman Addison Wesley, 1998., p. 116-118.
Rational expressions/functions
- Modeling variation/combined variation
 wt 
M  67.5 

 d 
2
What a nice way to show a modeling process, to compare
linear/quadratic/rational, and to combine them all into one
equation!
Problem from Advanced Algebra, University of Chicago School
Mathematics Project, Scott Foresman Addison Wesley, 1998., p. 116-118.
Rational Functions
Modeling
Suppose the average annual cost per year in electricity to run a refrigerator is
$92.
a.
Determine the total annual cost of a refrigerator costing $500 and lasting for
15 years.
b.
Determine the total annual cost of a refrigerator costing $500 and lasting for
20 years.
c.
Write a function describing the total annual cost C(n) of a $500 refrigerator
lasting n years.
d.
Graph your function.
e.
Determine any asymptotes and explain their meaning in terms of the
problem’s context.
f.
Suppose a refrigerator costs $1000. Is there an expected lifespan of the
refrigerator that would make it more cost-effective than the refrigerator
described in part a or part b?
Other questions: why look at annual cost versus total cost over the refrigerator’s
lifetime?
What other standards apply?
G.GPE.6
Find the point on a directed line segment between
two given points that partitions the segment in a
given ratio.
G.MG.4.MA
Use dimensional analysis for unit conversion to
confirm that expressions and equations make
sense.
G.SRT.1
Understand similarity in terms of similarity
transformations. Verify experimentally the properties
of dilations given by a center and a scale factor:
G.SRT.1a
A dilation takes a line not passing through the
center of the dilation to a parallel line, and
leaves a line passing through the center
unchanged.
G.SRT.1b
The dilation of a line segment is longer or shorter in
the ratio given by the scale factor.
What other standards apply?
F.TF.3 (+)
Use special triangles to determine
geometrically the values of sine, cosine,
tangent for pi/3, pi/4 and pi/6, and use the
unit circle to express the values of sine,
cosine, and tangent for pi - x, pi + x, and 2pi
- x in terms of their values for x, where x is
any real number.
G.SRT.6
Define trigonometric ratios and solve problems
involving right triangles. Understand that by
similarity, side ratios in right triangles are properties
of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
G.SRT.8
Use trigonometric ratios and the Pythagorean
theorem to solve right triangles in applied problems.
N.CN.3 (+)
Find the conjugate of a complex number; use
conjugates to find moduli and quotients of complex
numbers. Represent complex numbers and their
operations on the complex plane.
What other standards apply?
Any and all the standards on probability require an
ability to work with / understanding of fractions and
ratios.