The New Approach to High School Geometry
Required by Common Core
Tom Sallee
University of California, Davis
BE SURE TO ASK QUESTIONS
This talk is for you—not me.
Most basic geometric question
• What does it mean to say that two geometric
figures or objects are “the same”
Most basic geometric question
• What does it mean to say that two geometric
figures or objects are “the same”?
Generally it means that they are “congruent” or
at least “similar”.
So what does “congruence” mean?
• Discuss at your end of the table.
• I REALLY don’t want to hear theorems quoted.
Changes being required for geometry
courses.
Biggest one: rigid transformations are how you
determine congruence.
From CCSS-Math
First Geometry Standards on Congruence
• Experiment with transformations in the plane
• Understand congruence in terms of rigid
motions
Mathematical Agenda Today
• Topic 1 Look at relationships among various
congruence and similarity transformations.
• Topic 2 Use transformations to prove one or
two standard theorems.
Because this is not a formal math class
• I am not going to be precise about the
distinctions between a segment and the
measure of its length and ditto with angles. I
am trying to convey larger ideas and believe
you can translate the occasional imprecision
into what you need.
Basic Motions
• Reflection Across a Line
• Dilation With Respect to a Point
Reflections and Rigid Motions
Will talk about dilations and similarity later.
What is true about reflections
(across a line)?
Work with people to understand:
• What they are.
• What properties are preserved by reflections?
• What properties are not?
Postulate 1
• Every reflection preserves
linearity,
parallel lines,
angles
intersections
and lengths.
(and betweenness)
Other Basic Motions
Each of these is a product of reflections.
Translations
Rotations about a point
We will not be considering glide reflections.
Definitions
• A rigid motion or rigid transformation (or
congruence transformation) is the result of a
sequence of reflections, rotations, and
translations.
•
Two geometric figures are congruent if
there exists a congruence transformation
taking one exactly onto the other.
Theorem 1
• Every congruence transformation preserves
linearity, angles and lengths.
• We are simply saying that since linearity, etc.
is preserved at each step, it is preserved from
beginning to end.
What can we do with reflections?
• Key point: If you know what happens to a
single triangle in a congruence transformation
you know what happens to the entire space.
• For this course, we only care about
congruence and similarity transformations.
Congruence via reflections
• If ABC and A’B’C’ are congruent triangles, can
you find a single reflection that takes the point
A to the point A’?
Congruence via reflections
• If ABC and A’B’C’ are congruent triangles, can
you find a single reflection that takes A to A’?
• If ABC and AB’C’ are congruent triangles [note
A = A’ here, so the two triangles have a
common vertex], can you find a single
reflection that keeps A fixed and takes B to B’?
Congruence via reflections
• If ABC and A’B’C’ are congruent triangles, can you find
a single reflection that takes A to A’?
• If ABC and AB’C’ are congruent triangles [note A = A’
here, so the two triangles have a common vertex], can
you find a single reflection that keeps A fixed and takes
B to B’?
• If ABC and ABC’ are congruent triangles [note A = A’
and B = B’ here, so the two triangles have a common
edge], and C is not equal to C’, can you find a single
reflection that keeps A and B fixed and takes C to C’?
Theorem 3
Every congruence transformation is the product
of at most 3 reflections.
From reflections to other rigid motions
• .
What do reflections do to the plane?
• In particular,
are any points fixed by a reflection—i.e.
don’t move?
Points fixed by reflections
Are any points fixed by a reflection across a line?
• Answer: points are not moved by a reflection
if and only if they are on the line.
Reflections across intersecting lines
• Are any points fixed by two reflections (in
sequence) across intersecting lines?
Discuss
Reflections across intersecting lines
• Answer: a point is fixed by a sequence of two
reflections if and only if it is at the intersection
of the lines.
[One way is obvious; other way is also true
but less obvious.]
Only fixed point is intersection.
• Set up coordinate system so that first line of
reflection is the x-axis. Reflect first across it
preserves x-value. Reflecting then across any
other line will change x-value. Thus point will
not have been fixed.
So we get???
• A point is fixed by a sequence of two
reflections if and only if it is at the intersection
of the lines.
What kinds of rigid motions keep exactly one
point fixed?
Theorem 5
The product of two reflections across two lines
that intersect at C is a rotation about the center
C.
Reflections across parallel lines
• What do you think happens if the lines are
parallel?
Theorem 6
The result of two reflections across
parallel lines is a translation.
Theorem 6 (improved)
• Every translation is equivalent to two
reflections across parallel lines that are
perpendicular to the translation segment.
The distance between these lines is half the
translation distance.
• The easiest pair to describe probably is where
the first line is the perpendicular bisector of
the segment and the second one is through
the “proper” end of the segment.
Theorem 5 (improved)
• Every rotation is the product of two
reflections across lines that intersect at the
center of rotation. The angle between these
lines is half the rotation angle.
Essentially the same argument works. Again the
easiest pair to describe is probably the one that
takes the x-axis to where it will end up after the
rotation and then to reflect around that line.
New View of Congruence
A standard triangle congruence theorem takes
three (possibly separate) congruence transforms
and replaces them by a single one that implies
three other congruences.
SAS Congruence
• So SAS on the triangles ABC and DEF says that
if there is one congruence transform that
maps AB to DE, another congruence transform
that maps angle B to angle E, and a third
congruence transform that maps BC to EF,
then there exists a single congruence
transform that simultaneously takes A to D, B
to E and C to F thereby showing the
congruence of all corresponding sides and
corresponding angles.
Proof of SAS
• You would think we had done it earlier in
proof of Thm. 3, but we didn’t. There we
made the assumption that we had the
congruence of the conclusion, but that is what
we need to find.
Proof sketch of SAS
Idea is similar. Take the congruence
transformation that takes the rays that describe
angle ABC to angle DEF. Call the image A’B’C’.
Then B’ = E, and A’ lies along ray ED with B’A’
(that is, EA’) being congruent to BA and BA is
given congruent to ED. Hence, EA’ is congruent
to ED and thus, A’ = D. Similarly,
C’ = F, so the triangle A’B’C’ is the triangle DEF
and so ABC and DEF are congruent.
What could possibly go wrong?
Let’s see.
What could possibly go wrong?
Let’s see.
Take the congruence transformation that takes
the rays that describe angle ABC to angle DEF.
Call the image A’B’C’. Then B’ = E, and A’ lies
along ray ED.
• We need to know that A’ is on this ray and not
the other one. If A’ is on this ray, we still need
to know why C’ is on the other one.
What could go wrong-part 2?
• Suppose we know that A’ lies along ray ED
(and C’ lies on ray EF) with B’A’ (or EA’)
congruent to BA and BA is congruent to ED so
EA’ is congruent to ED
• At some point we need to establish the
transitivity of congruence.
What could go wrong-part 3?
• Then B’ = E, and A’ lies along ray ED with B’A’
(or EA’) congruent to BA and BA is congruent
to ED so EA’ is congruent to ED and thus, A’ =
D.
• We also need to establish that if Z lies on the
ray XY and that if XY is congruent to XZ, then
Y = Z.
So it is not as easy as we want.
• It appears that Euclid was correct when he said to
King Ptolemy, “There is no royal road to
geometry,” and rigid motions is not that road.
• The tradeoffs are among understandability, rigor,
and destroying belief in the value of proof.
[Why should a high schooler see the need to prove
that angle ABC is congruent to angle CBA? It is a
completely arbitrary exercise at this age.]
To be purely rigorous
We need to verify a lot of stuff that kids will not
care about.
Suppose all of these basic issues have been
dealt with, then we are in good shape.
Finally
Back to dilations.
Similarity transformations
• What is going to be the difference between a
congruence transformation and a similarity
transformation?
Discuss with someone near you.
What is true about dilations?
Work with people to understand:
• What they are.
• What properties are preserved by dilations?
• What properties are not?
Postulate 2
• Every dilation preserves
linearity,
parallel lines,
angles
intersections
and ratios of lengths.
(and betweenness)
Definition
• A similarity transformation is the result of a
sequence of reflections, rotations, translations
and dilations.
•
Two geometric figures are similar if …
what??
Theorem 2
• Every similarity transformation preserves
linearity, angles, and ratios of lengths.
Same idea as before.
How many dilations?
• If a similarity transform involves two dilations,
one that doubles distances and the other than
multiplies by 3, is there a single dilation that
can be used instead?
How many dilations?
• If a similarity transform involves two dilations, one that
doubles distances and the other than multiplies by 3, is there
a single dilation that can be used instead?
• So we can always content ourselves with just
worrying about a single dilation. (Basic
proof—you have a similarity transform, so
choose a triangle to transform and then look
at its ultimate result.)
Exploring similarity transforms.
• Suppose triangles DEF and D’E’F’ are similar,
what sequence of reflections and dilations
could show this?
• What would be your basic strategy?
State the theorem
• Theorem 4: Every similarity transformation
is the product of ….. what?
Theorem 4
Every similarity transformation is the product
of at most 3 reflections and one dilation.
AA and Side Ratios in Similar Triangles
• By the way we defined similar figures, AA
implies corresponding ratios of sides.
• What about the converse?
If two triangles have equal ratios of
corresponding sides, they are similar.
• Suppose ABC and A’B’C’ are two triangles such
that AB/A’B’ = AC/A’C’ = BC/B’C’. Then AAA is
true.
If two triangles have equal ratios of
corresponding sides, they are similar.
• Suppose ABC and A’B’C’ are two triangles such
that AB/A’B’ = AC/A’C’ = BC/B’C’.
• Note that we can conclude other equalities.
If AB/A’B’ = AC/A’C’, then AB/AC = A’B’/A’C’,
etc.
Sketch of proof
Take a similarity transform of ABC centered at A to
get A”B”C” so that A”C” is congruent to A’C’.
[Probably should argue we know how to do this.]
Then since by earlier argument, AC/A”C” = AB/A”B”
we know that AC = A”C”, etc. so triangle ABC is
congruent to triangle A”B”C” by SSS. Since ABC is
similar to A’B’C’ and angles are reserved with both
congruence transformations and similarity
transformations, we know corresponding angles are
equal.
Questions?
Before I move to a larger issue?
Key Question: What proofs are worth
doing? And what kinds of proofs?
Your thoughts? Discuss.
When are proofs worth doing?
• My answer would be when they discover or
validate something surprising. Examples:
•
•
•
•
•
Exterior Angle Theorem
Inscribed Angle Theorem for Circles
Law of Sines.
Area of Circle = 0.5*r*circumference
There are an infinite number of primes
The battle over what a proof is
• Must it be as rigorous as Euclid?
[With the clear understanding that Euclid
slipped up a lot on his basic assumptions, but
was very good after that.]
• What about demonstrations using Geogebra?
Are these proofs?
• What are we trying to do when we do proofs?
My thoughts
• We need to help everyone to learn to reason
clearly and deductively. Mathematics is
arguably the best place to do it—everyone
agrees on correctness of answers (if not
arguments).
• It is also the place to learn that your
assumptions decide your conclusions.
More thoughts
• Intuition is great. Most of science is done by
making conjectures and validating. But
people need to know that 3 or 10 examples
are not enough. But a 1000 are certainly
persuasive.
• I wish high school math did more with
programming—an even better place to learn
to reason clearly.