Transcript Patrick Slides may18

CCSS in Secondary Mathematics:
Changing Expectations
Patrick Callahan
Co-Director California Mathematics Project
Plan for this morning
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Changing expectations for Algebra
Do some algebra!
Changing expectations for Geometry
Do some geometry!
The course titles may be the same, but
the course content is not!
Common Core Algebra and Geometry are quite
different than previous CA Algebra and
Geometry courses!
Conrad Wolfram’s TED Talk:
What is math?
1. Posing the right questions
2. Real world  math formulation
3. Computation
4. Math formulation  real world, verification
Conrad Wolfram’s TED Talk:
What is math?
1. Posing the right questions
2. Real world  math formulation
3. Computation
4. Math formulation  real world, verification
Humans are vastly better than
computers at three of these.
Conrad Wolfram’s TED Talk:
What is math?
1. Posing the right questions
2. Real world  math formulation
3. Computation
4. Math formulation  real world, verification
Yet, we spend 80% or more of math
instruction on the one that computers
can do better than humans
Conrad Wolfram’s TED Talk:
What is math?
1. Posing the right questions
2. Real world  math formulation
3. Computation
4. Math formulation  real world, verification
Note: The
CCSS would indetify Wolfram’s description
of math to be Mathematical Modeling, one of the
Mathematical Practices that should be emphasized
K-12.
Sample Algebra Worksheet
This should look familiar.
What do you notice?
What is the mathematical
goal?
What is the expectation of
the student?
A sample Algebra Exam
A sample Algebra Exam
I typed #16 into
Mathematica
Look at the circled answers.
What do you notice?
Algebra ≠ Bag of Tricks
To avoid the common experience of
algebra of a “bag of tricks and
procedures” we adopted a cycle of
algebra structure based on a family of
functions approach.
HS Algebra Families of Function Cycle
CONTEXTS
FUNCTIONS
(modeling)
EQUATIONS
(solving,
manipulations
Families of Functions:
Linear (one variable)
Linear (two variables)
Quadratic
Polynomial and Rational
Exponential
Trigonometric
ABSTRACTION
(structure,
precision)
Context
From Dan Meyer’s blog
Model with functions
Equations
g(x)  2.8x 2  2.43x  3.77
0  2.8x 2  2.43x  3.77
You can’t “solve” a function. But functions can be analyzed
and lead to equations, which can be solved.
What was the maximum height of the ball?
How close did the ball get to the hoop?
Symbolizing,
manipulating,
Equivalence…
Abstracting (structure, generalization)
Examples:
The maximum or minimum occurs at the midpoint of the roots.
The sign of the a coefficient determines whether the parabola is
up or down (convexity)
The c coefficient is the sum of the roots.
The roots can be determined in multiple ways: quadratic
formula, factoring, completing the square, etc.
(x  p)(x  q)  x 2  2( p  q)x  pq
b  b 2  4ac
ax  bx  c  0  x 
2a
2
HS Algebra Families of Function Cycle
CONTEXTS
FUNCTIONS
(modeling)
EQUATIONS
(solving,
manipulations
Families of Functions:
Linear (one variable)
Linear (two variables)
Quadratic
Polynomial and Rational
Exponential
Trigonometric
ABSTRACTION
(structure,
precision)
Algebra
Area
A=
W
Perimeter
L
P=
Algebra
Area
A = LW
W
Perimeter
L
P = 2(L+W)
Can you find a rectangle such that the
perimeter and area are the same?
?
The name “Golden
Rectangle” was taken,
So let’s call such a
rectangle a “Silver
Rectangle”
“Silver Rectangles”
Area = 16
Perimeter = 16
4
4
Silver square symbolic solution
A  LW  x * x  x 2
P  2(L  W )  2(x  x)  4 x
4
AP:
x2  4x  x  4
4

Other silver rectangles
2k
k
A  LW  k * 2k  2x 2
P  2(L  W )  2(k  2k)  6k
AP:
2k 2  6k  k  3

Algebra outside the box
Volume
V=?
Surface Area
S=?
H
W
L
Edge length
E=?
Algebra outside the box
Volume
V = LWH
Surface Area
S = 2(LW+HW+LH)
H
W
L
Edge length
E = 4(L+W+H)
BONUS QUESTION:
Can you find a “Silver Rectangular Prism” (aka Box)?
Volume
V = LWH
Surface Area
S = 2(LW+HW+LH)
H
W
L
Can you find a box with V=S=E?
Edge length
E = 4(L+W+H)
Geometry
Why geometric transformations?
NAEP item examples…
The 2007 8th grade NAEP item below was classified as “Use similarity of right
triangles to solve the problem.”
Why is this so difficult?
The 2007 8th grade NAEP item below was classified as “Use similarity of
right triangles to solve the problem.”
Only 1% of students answered this item correctly.
The 1992 12th grade NAEP item below was classified as
“Find the side length given similar triangles.”
The 1992 12th grade NAEP item below was classified as
“Find the side length given similar triangles.”
Only 24% of high school seniors answered this item correctly.
Why are these items so challenging?
Are these “the same”?
Are these “the same”?
Are these “the same”?
Are these “the same”?
Precision of meaning (or lack thereof)
Much of mathematics involves making ideas precise.
The example at hand is the challenge of making precise the concept
of Geometric Equivalence.
There is some common sense notion of “shape” and “size”.
Same “shape” and same “size”
(“CONGRUENT”)
Same “shape” and different “size”
(“SIMILAR”)
In a survey of 48 middle school teachers, 85% gave these definitions
Are these “congruent”?
Well, they seem to have the same shape and same size.
But one is …”upside down”… “pointing a different way”…
“they are the same but different”
If we think these are geometrically equivalent/congruent,
then we are implicitly ignoring where and how they are
positioned in space. We are allowed to “move things
around”
More precision needed…
The main problem with the definition
“same shape, same size”
is “shape” and “size” are not precise
mathematical terms
Congruence and Similarity
Typical (High School) textbook definitions:
Pg 233: Figures are congruent if all pairs of corresponding
sides angles are congruent and all pairs of corresponding
sides are congruent.
Pg 30: segments that have the same length are called congruent.
Pg 36: two angles are congruent if they have the same measure.
Pg 365: Two polygons are similar polygons if corresponding
angles are congruent and corresponding side lengths are
proportional.
Another implicit problem…
Typical textbook definitions:
Pg 233: Figures are congruent if all pairs of corresponding
sides angles are congruent and all pairs of corresponding
sides are congruent.
Pg 30: segments that have the same length are called congruent.
Pg 36: two angles are congruent if they have the same measure.
Pg 365: Two polygons are similar polygons if corresponding
angles are congruent and corresponding side lengths are
proportional.
What does “corresponding” mean?
“Correspondence” causing problems?
The 1992 12th grade NAEP item below was classified as
“Find the side length given similar triangles.”
Geometric Transformations
An alternate approach to congruence and similarity is using geometric
transformations (1 to 1 mappings of the plane). An isometry is a
transformation that preserves lengths.
Definition: Two figures are congruent if there is an isometry mapping one to
the other.
The Common Core State Standards puts it this way:
CCSS 8G2: Understand that a two-dimensional figure is congruent to
another if the second can be obtained from the first by a sequence of
rotations, reflections, and translations
CCSS 8G4: Understand that a two-dimensional figure is similar to
another if the second can be obtained from the first by a sequence of
rotations, reflections, translations, and dilations
The 1992 12th grade NAEP item below was classified as “Find the side length given similar
triangles.”
8
A rotation and a
dilation show the
corresponding sides
of the similar
triangles.
5
6
8
5
6
8
x
12.8
Figure A
Figure B
Recall, only 24% of high school seniors answered this item correctly.
I conjecture that the students didn’t see the correspondence, hence set up the
problem incorrectly, e.g. 6/8 = 5/x .
Simple example: Vertical Angle Theorem
mA  mB  180
mC  mB  180

mA  mC  0

mA  mC

Why the new approach?
We have been using the old “lengths and angles” or
“shape and size” approach and it has been working fine.
Why change to this new “transformations approach?
First, length and angles restricts to polygonal figures.
What about curves? Circles?
Second, geometry education is not “working fine” (NAEP)
Third, transformations are not new.
Euclid (c. 300 BC)
Euclid’s Common Notions
1) Things which are equal to the same thing are also equal to one another.
2) If equals be added to equals, the wholes are equal.
3) If equals be subtracted from equals, the remainders are equal.
4) Things which coincide with one another are equal to one another.
5) The whole is greater than the part.
Interestingly, “congruence” does not appear anywhere in
Euclid’s elements.
Euclid implicitly uses “superposition”
Book 1, Proposition 4.
If two triangles have the two sides equal to two sides
respectively, and have the angles contained by the
equal straight lines equal, they will also have the base
equal to the base, the triangle will be equal to the
triangle, and the remaining angles will be equal to the
remaining angles respectively, namely those which the
equal sides subtend.
Proof: …For, if the triangle ABC be applied to the triangle
DEF, and if the point A be placed on the point D and the
straight line AB on DE, then the point B will also
coincide with E, because AB is equal to DE…
Klein’s Erlangen Program of 1872
Felix Klein (18491925)
Geometry is the study
of properties of a
space that are
invariant under a
group of
transformations.
Why Transformations?
1. Historical
2. Symmetry
3. Embodied
Symmetry
Symmetry
"All of mathematics is the study of symmetry,
or how to change a thing without really
changing it."
H.S.M. Coxeter
translations
reflection
Embodied
The fact that we exist and interact in 3dimensional space with our bodies has
been posited as deeply impacting our
cognition.
We physically
experience rotations,
translations,
reflections, and
scaling all the time.
Why transformations?
“[Transformations] give a unifying concept to the geometry course.
Traditional geometry courses have unifying concepts – set, proof –
but these are not geometric in nature. The concept of
transformation, essential to a mathematical characterization of
congruence, symmetry, or similarity, and useful for deducing
properties of figures is indeed a unifying concept for geometry.”
Coxford and Usiskin, Geometry a Transformational Approach (1971)
Non-transformational approach
"Other books on geometry often refer to equal triangles as "congruent"
triangles. They do this to indicate not only that corresponding sides and
angles are equal, but also that this equality can be shown by moving
one triangle and fitting it on the other. They define "congruent" in
terms of the undefined ideas of "move" and "fit". The logical
foundation of our geometry is independent of any idea of motion.”
"Later, when we wish to link our geometry with problems of the
physical world about us, we shall simply take as undefined the idea of
motion of figures (without change of shape or size).”
Birkhoff and Beatley, Basic Geometry (1932)
The CCSS does both
Students are expected to know that the traditional
definition is equivalent to the transformational definitions
for congruence and similarity.
G-CO 7
Use the definition of congruence in terms of rigid motions to show
that two triangles are congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are congruent.
G-SRT 2
Given two figures, use the definition of similarity in terms of
similarity transformations to decide if they are similar; explain using
similarity transformations the meaning of similarity for triangles as
the equality of all corresponding pairs of angles and the
proportionality of all corresponding pairs of sides.
Where do transformations connect to other parts of
mathematics?
Graphing functions
Translations:
f (x)  f (x  h)  k
What about the
“general form” of a
trig function?
Asin( Bx  C)  D
Transformations are functions
We can use coordinates to express the transformations in function notation.
To translate a point by a fixed vector (a,b):
T(x, y)  (x  a, y  b)
A dilation centered at the origin with scale factor k:

D(x, y)  (kx,ky)
Are parabolas similar?
Similar parabolas
f (x)  ax  bx  c
2
f (x)  k(x  p)  q
2
f (x) ~ kx
f (x) ~ x
2
2
Geometric Transformations and Complex Numbers
Transformation
Complex algebra
translation
addition
p p  z
rotation
multiplication
dilation
multiplication
p p z
reflection
conjugation

Multiplying by Z

Dilate by the length
of Z
and
Rotate by the angle
of Z

(both centered at 0)
formula
p p