Patrick Callahan
Co-Director California Mathematics Project

• 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!

What is math?
1. Posing the right questions
2. Real world  math formulation
3. Computation
4. Math formulation  real world, verification

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.

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

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.

This should look familiar.
What do you notice?
What is the mathematical goal?
What is the expectation of the student?

I typed #16 into
Mathematica

Look at the circled answers.
What do you notice?

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
ABSTRACTION
(structure, precision)
Families of Functions:
 Linear (one variable)
 Linear (two variables)
 Quadratic
 Polynomial and Rational
 Exponential
 Trigonometric

From Dan Meyer’s blog


g ( x )
 
2.8
x
2 
2.43
x

3.77
0
 
2.8
x
2 
2.43
x

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 ax
2
 bx
 c

0
 x

 b
 b
2

4 ac
2 a


HS Algebra Families of Function Cycle
CONTEXTS FUNCTIONS
(modeling)
EQUATIONS
(solving, manipulations
ABSTRACTION
(structure, precision)
Families of Functions:
 Linear (one variable)
 Linear (two variables)
 Quadratic
 Polynomial and Rational
 Exponential
 Trigonometric

L
W
Area
A =
Perimeter
P =

L
W
Area
A = LW
Perimeter
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”

4
Area = 16
Perimeter = 16
4

4

4
A

LW
 x * x
 x
2
P

2( L

W )

2( x
 x )

4 x
A

P : x
2

4 x
 x

4

k

2k
A

LW
 k * 2 k

2 x
2
P

2( L

W )

2( k

2 k )

6 k
A

P :
2 k
2

6 k
 k

3

H
L
W
Volume
V = ?
Surface Area
S = ?
Edge length
E = ?

H
L
W
Volume
V = LWH
Surface Area
S = 2(LW+HW+LH)
Edge length
E = 4(L+W+H)

BONUS QUESTION:
Can you find a “Silver Rectangular Prism” (aka Box)?
H
W
L
Can you find a box with V=S=E?
Volume
V = LWH
Surface Area
S = 2(LW+HW+LH)
Edge length
E = 4(L+W+H)

The 2007 8th grade NAEP item below was classified as “Use similarity of right triangles to solve the problem.”

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 12 th grade NAEP item below was classified as
“Find the side length given similar triangles.
”

The 1992 12 th 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?

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

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”

The main problem with the definition
“same shape, same size” is “ shape ” and “ size ” are not precise mathematical terms

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.

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 12 th grade NAEP item below was classified as
“Find the side length given similar triangles.
”

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 12 th 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.
6 5
8
5
8
6 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


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.

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…

Felix Klein (1849-
1925)
Geometry is the study of properties of a space that are invariant under a group of
transformations.

"All of mathematics is the study of symmetry, or how to change a thing without really changing it."
H.S.M. Coxeter translations reflection

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.

“[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)

"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)

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?

Translations: f ( x )
 f ( x
 h )
 k
What about the
“general form” of a trig function?
A sin( Bx

C )

D



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 )


f ( x )
 ax
2  bx
 c f ( x )
 k ( x
 p )
2  q f ( x ) ~ kx
2 f ( x ) ~ x
2

Geometric Transformations and Complex Numbers
Transformation translation rotation dilation reflection
Multiplying by Z
Dilate by the length of Z and
Rotate by the angle of Z
(both centered at 0)
Complex algebra formula addition p
 p
 z multiplication multiplication p
 p
 z p
 p

