CCSS in Secondary Mathematics: Changing Expectations

CCSS in Secondary Mathematics:Changing Expectations

Patrick CallahanCo-Director California Mathematics Project

Plan for this morning

• 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 questions2. Real world math formulation3. Computation4. Math formulation real world, verification



Humans are vastly better than computers at three of these.



Yet, we spend 80% or more of math instruction on the one that computers can do better than humans



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

ABSTRACTION(structure, precision)

Families of Functions: Linear (one variable) Linear (two variables) Quadratic Polynomial and Rational Exponential Trigonometric

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

ax 2 + bx + c = 0 ⇔ x =−b ± b2 − 4ac

2a

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

Algebra

L

W

Area

A =

Perimeter

P =

Algebra

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”

“Silver Rectangles”

4

4

Area = 16Perimeter = 16

Silver square symbolic solution

4

4

€

A = LW = x * x = x 2

P = 2(L +W ) = 2(x + x) = 4x

A = P :

x 2 = 4x ⇒ x = 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 = ?

Edge length

E = ?L

W

H

Algebra outside the box

Volume

V = LWH

Surface Area

S = 2(LW+HW+LH)

Edge length

E = 4(L+W+H)L

W

H

BONUS QUESTION:Can you find a “Silver Rectangular Prism” (aka Box)?

Volume

V = LWH

Surface Area

S = 2(LW+HW+LH)

Edge length

E = 4(L+W+H)L

W

H

Can you find a box with V=S=E?

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.”


Only 24% of high school seniors answered this item correctly.

Why are these items so challenging?

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?


Geometric TransformationsAn 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


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 .

8

6 5

12.8

x88

65

Figure BFigure A

A rotation and a dilation show the corresponding sides of the similar triangles.

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 Notions1) 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 (1849-1925)

Geometry is the study of properties of a space that are invariant under a group of transformations.

Why Transformations?

1. Historical2. Symmetry3. 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

reflection

translations

EmbodiedThe fact that we exist and interact in 3-dimensional 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

G-CO 7Use 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.

Students are expected to know that the traditional definition is equivalent to the transformational definitions for congruence and similarity.

Where do transformations connect to other parts of mathematics?

Graphing functions

€

f (x) → f (x + h) − k

Asin(Bx + C) + D

Translations:

What about the “general form” of a trig function?

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 2 + bx + c

f (x) = k(x − p)2 + q

f (x) ~ kx 2

f (x) ~ x 2

Geometric Transformations and Complex Numbers

Transformation Complex algebra formula

translation addition

rotation multiplication

dilation multiplication

reflection conjugation€

p → p + z

€

p → p • z

€

p → p

Multiplying by Z

Dilate by the length of Z

andRotate by the angle of Z

(both centered at 0)

Documents

CCSS in Secondary Mathematics: Changing Expectations