The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics » geometry.pre-college

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Conversation Shared (Beware: Long Message!)
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List  
Keith Grove

Posts: 70
Registered: 12/3/04
Conversation Shared (Beware: Long Message!)
Posted: Mar 12, 1994 1:44 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

March 1, 1994


I've mailed the old lessons and the new geometry outline. Here are 7 lessons
I've written in the last couple of weeks. Your comments and suggestions are
requested. You should be advised that my underlying goal for the course is
to maximize the acquisition of geometric knowledge-a traditional,
essentialist point of view. I am well aware that this position is
currently out of favor with the NCTM, but too many of my bright students
have thanked me for the content based course I offer for me to give it up.
On the other hand, where connections can honestly be made with the "world"
such connections should be made, however, in my view, a problem based
course leaves too many content holes.

I look forward to your comments. Sorry that email does not transmit diagrams
and text is sometimes broken up. At some point I will send hard copy.

March 2, 1994


Philosophy aside for a moment - I work in a district where the parents
DEMAND content. 98% of the student go to 4 year colleges and SAT rule the
day. So our work together may find congruent goals.

Could you say more about what you mean by your "bright" students? My
experience and my childhood indoctrination tell me that certain students
seem to "get it" faster that others. BUT as I read Howard Gardner's book
"The Unschooled Mind" and notice that the "basis" level students seems to
predominately come from lower economic backgrounds and more emotionally
unbalanced homes - I wonder. As then I say to myself - what if I'm wrong
in putting kids into categories ... and why not assume that all kids can -
and work like heck to find ways for them to succeed - through varying
teaching methods and assessment, etc.

I would be interested to know how you think about such things!:-)

I believe we will have a few snow days this weeks - I hope to find time to
read your lessons then - thanks alot for email them. And I received a
thick package in the mail yesterday - my summer reading!!

>Sorry that email does not transmit diagrams and text is sometimes broken up.
>At some point I will send hard copy.

Do you have Sketchpad? And are you aware that sketches can be sent as
"attached documents"?

Talk with you soon.

March 3, 1994


>>>>"Bright kids" is a description of demonstrated ability to learn quickly. It
>>is not a static categorization, nor a comment on potential or future
>>achievement. I consider it a professional obligation to provide
>>opportunities for the quick learners to develop their abilities. I also have
>>a lot of
>>respect for, and devote a lot of my free time to, hard working students who
>>often rise to the challenge. I do not categorize by group status, however I
>>do respond to demonstrated behaviors and attitudes.

>>I have friends today who were my students over 15 years ago. They came from
>>inner city homes, but they had drive, a willingness to work hard, and good
>>brains. From a school where many kids were at the 16th percentile, these
>>students bent on to college, did well, and are professional people today.
>>You might refer to the March,1980 issue of the MT to read an article I wrote,
>>"The Neglected Minority." It says a lot about my educational philosophy
>>and goals.

>Myron -

March 4, 1994


>I sense that I may have offended you - or at least put you on the defensive.
>It's obvious where my biases lie in regard to this issue, but I hope that
>it is equally obvious that I'm struggling to find the common ground between
>what my experiences tell me and what I've read in recent years. I will try
>to get my hands on the March, 1980 issue of MT - I'm sure I will find it
>Perhaps the issue for me is more one of language. When I hear bright - I
>think that the opposite is dull. When I hear smart - I think the opposite
>is dumb.
>It's clear from your response that this is not how you were using the
>language and again I apologize if my words indicated such a judgement.

>Perhaps it is different where you work, but in the high school where I live
>- when most teachers refer to the bright students - they are saying that
>some kids have more intelligence than others -- and even more perplexing to
>me they are saying that we adult teacher can tell which ones have how much.
> At the same time, most of these teacher use the traditional lecture
>approach amost exclusively - and when students don't "get it" - they are
>"slow" or "dull" or "the dumbies".
>Now that I have a little son - I am especially in tune with this issue. I
>hope that no teacher ever calls my son bright or not so bright. Its
>obvious to me that he is so much more than this or any other category.
>And finally, I should admit that I work at a predominately white,
>upper-middle class suburban school. The minority students I see are amost
>exclusive Metco students (bussed from Boston). Of
>the thirty some student in our high school - I would say that about 27 of
>them are in the lower track. When I ask them about their previous
>schooling - I can't believe the disparities between that experience and
>mine. When I ask them about where they live - they talk about not feeling
>safe to go out after dark. I'm sure that I'm not telling you anything you
>don't already know.
>Once again - sorry and I hope no damage was done.
>I'm going to be working on a rough draft of a outline which will use Mosaic
>as an interface for a joint student teacher on-line geometry book. If
>you're interested please offer your feedback. Stay tuned.

March 5, 1994


No apology needed. It has been my experience that, "intelligence" aside,
there is no substitute for hard work on the part of the students. Indeed,
the hard worker sometimes out perform the lazy student with more "inate
ability." I become defensive when children's low performance is stated as
a problem, and then solutions involve changes in curriculum, teaching styles,
more money, society, everything but the student's failure to assume
responsibility for his/her counterproductive behavior. I look forward to
discussing educational issues with you.

Question:Within the "bad" schools of Boston, my home town, I suspect there
are a some highly motivated bright kids. Are they the ones moved to
schools, like yours, where they can mix with kids with similar ambitions?
Where they can receive an opportunity to develop their mental abilities?
Is there any advantage to sending unmotivated students who do not study to
"better" schools? Indeed, to a large degree, isn't it the quality of the
students that determines if a school is "good" or "bad?"

The Mosaic idea is an exciting one. Unfortunately, I know so little about
how Mosaic works that I can't be much help. I have it, thanks to Annie, Dave
and Steve, and I see its potential, but I do not know how to produce the page.
But an interactive, changeable "text" would be very useful to those able to
access it.

I'll be sending more lessons soon.

March 6, 1994


Good questions. All I can tell you is that of the 20 or so Metco students
that I have taught - over 80% have not been highly motivated. They seem to
accept failure or at least below average performance. Very perplexing for
me - I really don't know how they are chosen. The best that I can say is
that I believe I really don't have a clue as to what is going on with the
Metco population. My life experiences and culture is too different.

>The Mosaic idea is an exciting one. Unfortunately, I know so little about
>how Mosaic works that I can't be much help. I have it, thanks to Annie, Dave
>and Steve, and I see its potential, but I do not know how to produce the page.
>But an interactive, changeable "text" would be very useful to those able to
>access it.

Well - I don't know much about it either. Steve has set up a framework for
the interactive, changeable "text". I haven't even gotten it yet. He
posted it today. I'll forward the message to you under separate cover.

To access: Under the File Command (I think - the first choice on the left)
select Open URL and then type
hhtp:// in the URL long
narrow rectange. (The last letter in the hhtp is a "L" not a "one")

>I'll be sending more lessons soon.

Great - I must admit that I'm swamped with work (so what's new for us
internet surfers hey?), but I will get to them. This time of year I find
myself thinking a lot about how I will start next year - does that happen
to you or am I wierd in this way?

What do you think about sharing our conversations on the
geometry.pre-collge newsgroup? It seems to me that these "types" of
conversations are missing. I am fortunate to be having them regularly
thanks to the existence of the Forum. But no matter how hard I have tried
(not very hard lately) I can't seem to get them going on the
geometry.pre-college newsgroup.

Myron's Lesson Plans:

OBJECTIVE: Be able to list 7 basic geometric parts

( What are some basic geometric terms? )

Activity: Brainstorm, putting responses on board

Notes for journal:
point A. or P1.
? (geometric representation for basic terms)
Assessment: Unscramble and draw the geometric representation

1. epnla
2. ptnio
3. nlei eegsmtn
4. aepsc
5. aelgn
6. lein
7. ayr

Objective: Be able to define and name a point

( What do you know about points? naming? size? )

Activity: Brainstorm- put responses on board

( Use Sketchpad to put points on screen )

( How are points named? )

( What is the size of a point? )

Notes for journal:
a. points are named using capital letters
b. points have no dimensions
c. a point is a location in space
d. point is an undefined term in Euclidean Geometry
e. Euclid, Greek mathematician, 2300 years ago,
organized geometric knowledge in 13 books
known as the Elements.

1. List 7 basic terms of Euclidean Geometry
2. Locate and name 5 locations on a paper
3. A point has how many dimensions?
4. Define point.

Objective: Be able to find the distance between 2 points on
a horizontal or vertical number line

Activity: Use Explorer software coordinate graph to place
points on x axis , name points, give coordinates,
take predictions for distance between 2 points. Show
computer answer. Do several times.

( How can the distance be predicted using the
x coordinates? )
Brainstorm and put results on board.

Repeat using the y axis. Use the ordinates to predict
the distance between two points. Show computer

( How can the distance between points on a vertical
segment be determined? )
Brainstorm and put results on board.

Notes for journal:
a. Distance between two points on a horizontal
segment can be determined by the absolute
value of the difference of the x coordinates.
b. Distance between two points on a vertical
segment can be determined by the absolute
value of the difference of the y coordinates.
c. Distance between points is always a positive
number. (or zero if two points are the same.)

Assessment: Find the distance between the following points:
1. (4,0) and (-5,0) 2. (0,4) and (0, -5) 3. (6,0)
and (-6,0)
4. Find two points on the x-axis 6 units from ( 4,0).
5. Find two points on the y axis 4 units from b (0,1).
6. State how to find the distance between two points on a
horizontal or vertical segment.

Objective: Be able to find the length of an oblique segment
(distance between two points in a plane)

Activity: Use Explorer software to plot points that are integral
measures of hypotenuse of right triangle; predict
length and then confirm with computer measurement.
( How am I doing it? )

Make table as follows:

AB BC AB squared BC squared AC
3 4
5 12
7 24
8 15
9 40
11 60
12 35
16 63
20 21

Brainstorm how AC is related to AB and BC.

Notes for journal:
a. In a right triangle, the sum of the square of the legs
equals the square of the hypotenuse. (Pythagorean Theorem)
b. To find the length of an oblique segment, use the
Pythagorean Theorem. The horizontal leg is the length
of a horizontal segment, | x2 -x1|; the vertical leg
is the length of a vertical segment, | y2 - y1 |; The
length of the segment is the square root of the sum
of the square of the
c. "Good Guys" to be memorized: 3,4,5; 5,12,13; 7,24,25;
8,15,17; 9,40,41; 11,60,61; 12,35, 37; 16,63,65;

Assessment: Find length of the segments with following endpoints.
1. ( 2,6) (-1,10) 2. (-2,-5) (3,7) 3.
(3,-7) (-4,17)
4. (4,8) (-5,48) 5. ( 2,9) (-6,24) 6.(-3,5)
7. ( -3,-5) (9,30) 8. (5,9) (-11,72) 9. ( 4,9)
10. (2,5) (3,4) 11. (3,0) ( 5,-3)

Objective: Be able to name and define a line

( What do you know about lines? naming? dimensions? length? )

Activity: Use sketchpad to put several lines on the screen.
Brainstorm answers to following questions.

( How are the lines named? )
( How wide is a line? How many dimensions? )
( How long is a line? )
( How many lines through 1 point? 2 points? )
( How would you define a line? )

Notes for journal:
a. Lines may be named by a single lower case letter or
by naming any two points on the line and placing the
symbol for a line above them.
b. A line has 1 dimension, length.
c. A line has no endpoints and has infinite length.
d. An infinite number of lines can be drawn through
1 point; Exactly one line can be drawn through
2 points.
e. Line is the second undefined term in Euclidean


Objective: Be able to identify and define types of lines

Activity: Put lines on screen, one at a time, using
Sketchpad, and ask students to brainstorm
observations for each one.
For oblique lines, use Explorer, to show relationship
between y and x coordinates.

a. horizontal line / vertical line / oblique line
b. Intersecting lines
c. parallel lines
d. perpendicular lines
e. skew lines (use arms)

Notes for journal:
a. A line with all the same y coordinates (ordinates) is a
horizontal line. The line has no verical change and so
has a slope of zero. Equation is y = a number.
b. A line with all the same x coordinates (abscissa) is a
vertical line. The line has no slope since the change
in the horizontal is zero, and division by zero is
The equation of a vertical line is x = a number.
c. A line that is neither horizontal nor vertical is oblique.
Slope is positive if y coordinate increases as x coordinate
increases; slope is negative if y coordinate decreases as
x coordinate increases. Equation is y = mx + b where "m"
is the slope; b is the y-intercept; x and y are
of points on the line.
d. Intersecting lines have 1 point in common. If the
angle formed by the intersecting lines is a right
angle (90°)
then the intersecting lines are perpendicular. Intersecting
lines have different slopes; perpendicular lines that are
not horizontal/vertical have slopes whose product is -1
(opposite reciprocals)
e. Lines that are coplanar and do not intersect are parallel.
Parallel lines have the same slope.
f. Skew lines are lines that cannot be contained on a flat
surface (plane).

1. What type line contains (4,7) (12,7)? (4,0) (4,-2)?
(4,1) (3,7)?
2. What kind of lines have a point in common?
3. What kind of lines form right angles?
4. What kind of lines do not intersect?
5. Find distance between: (3,0) (-4,0); (5,1) (5,7);
(2,5) (-3,17);

Objective: Be able to plot points on cartesian plane and predict
image after reflection

Activity: Use Explorer grid to plot and reflect points over x-axis;
y-axis; origin; and y=x line. Take predictions prior to
showing result on screen. Do many for each reflection.

Aim for generalizations:
a. image of (x,y) after reflection over x axis is ...
b. image of (x,y) after reflection in y axis is ...
c. image of (x,y) after reflection through origin is...
which is equivalent to rotating through ____
d. image of (x,y) after reflection in y=x line is ...

Notes for journal:
a. (x,y) becomes (x,-y) when erflected in x-axis.
b. (x,y) becomes (-x,y) when reflected in y axis.
c. (x,y) becomes (-x,-y) when reflected through the origin.
d. (x,y) becomes (y,x) when reflected in y=x line.
e. The cartesian coordinate plane was invented by Descartes
about 350 years ago.

a. Plot the following on the cartesian plane:
A(4,0) B(0,6) C(-2,4) D(-3,-5) E(4,-2) F(0,0)
b. What is the image of (2,3) when reflected in x-axis?
c. What is image of (-4,5) when reflected in y-axis?
d. What is the image of (2,5) when reflected in y=x line?
e. What is the image of (-2,-3) when reflected in origin?
f. What is the distance from (5,6) to (9,6)?
g. What is the distance from (3,7) to (3,-10)?
h. What is the distance from (-2,-6) to (3,6)?
i. Locate two point on the x-axis 5 units from (1,0).
j. Locate two points on the y-axis 3 units from (0,0).
k. Name 2 of the 3 undefined terms in Euclidean geometry.
l. A line has how many endpoints? dimensions?
m. List 9 "good guys" (Pythagorean triples).
n. (3,4) & (4,4) lie on a _________ line; (3,4) & (3,7) lie
on a ________ line; (4,-1) & (2,5) lie on a _______ line.

Myron Goldman (
(215) 635 1332
Dear Internet Readers,

The following "laundry list" has been translated by Steve Weimar in a
Mosaic Page.

To access: Under the File Command (I think - the first choice on the left)
select Open URL and then type
hhtp:// in the URL long
narrow rectange. (The last letter in the hhtp is a "L" not a "one")

Philosophy of Using Mosaic Format and of Interactive Nature of
this on-line book:
-live the question
-democratic knowledge
-idea of democracy being about creating - not simply choice
-use of consensus as a decision making model

-present a structure and format for the creation of an interactive
geometry experience (a book without a binder)
-capable of providing an Internet user with access to the lastest
software, research, pedagogies, and resources
-containing student and teacher sketches and journals
-linking teachers and students in current research projects -jointly
developed and implimented
-be a resource for any Internet user who wishes to "discover" the geometry
of his or her life
-to begin to address the questions: Geometry for What? How can Geometry
help students better understand themselves and their world? Are there
social, economic, and policial issues that an understanding of geometry
could help student better undersand?

-how students and teachers can join or enter the process

Evaluation of the on-going effort:
-philosophy of evaluation
-evaluation documents
-everyone could share his/her reflections of the current product
and process
-offer new directions, etc.

-portfolio of various assessment models
-include traditional applications (SAT type)
-include project orientation, portfolios, authentic exhibitions
-digital portfolios

-could access students' understandings of the topics (parallelism,
-include student sketches and journals
-include those areas which could be viewed through a particular
topic lens (for instance viewing orienteering through parallelism or
relationships among family member through triangles)
-provide definitions and diagrams

"Real World":
-approach the topics through the activity or profession (for
instance through architecture, carpentry, orienteering, psychology,
pragmatics of human communication, social responsibility, or the population
-approach the topics through the other traditional school
disciplines: science, social studies, art, etc.)

Geometry Relationships (Conjecture):
-sketchpad diagrams of various relationships
-completed by students, teachers, and others
-their applications to the world (ie. pictures of bridges or buildings)

-software and instructions for its use
-texts and lesson plans (including the reflections of teachers and
students who have used them)

-report of completed research
-current research possibilities (students and teacher could design their
own research projects - ie. student and teacher as researcher)

-history of geometry/mathematics
-important contributors - with their pictures of course


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.