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Unit 1, Activity 2, Extending Number and Picture Patterns
Geometry
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
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Unit 1, Activity 2, Extending Number and Picture Patterns
Date ___________
Name ________________________
Extending Patterns and Sequences
When presented with a sequence and asked to find the next term, inductive reasoning is applied.
Analyze the specific examples provided, determine a pattern, and then find the missing term.
Making a prediction about missing terms is called making a conjecture.
Examples: For each of the following, write the next two terms and describe the pattern.
1) 2, 4, 6, 8, 10, … _____, _____
2) -1, 0, 1, 2, 3, … _____, _____
3) 4, 7, 10, 13, 16, … _____, _____
4) 1, 4, 9, 16, 25, … _____, _____
5) 1, 3, 6, 10, 15, … _____. _____
6) 1, 3, 7, 15, 31, 63, … _____, _____
7) 1, 1, 2, 3, 5, 8, … _____, _____
8) 3, 5, 9, 15, 23, … _____, _____
Inductive reasoning can also be used to find missing terms in sequences and patterns dealing
with pictures. Draw the next two figures for each of the following and describe the pattern.
9)
10)
11)
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 1
Unit 1, Activity 2, Extending Number and Picture Patterns with Answers
Date ___________
Name ________________________
Extending Patterns and Sequences
When presented with a sequence and asked to find the next term, inductive reasoning is applied.
Analyze the specific examples provided, determine a pattern, and then find the missing term.
Making a prediction about missing terms is called making a conjecture.
Examples: For each of the following, write the next two terms and describe the pattern.
1) 2, 4, 6, 8, 10, … __12_, _14__
even numbers or +2
2) -1, 0, 1, 2, 3, … __4__, __5__
add 1 to each
3) 4, 7, 10, 13, 16, … _19_, _22_
add 3
4) 1, 4, 9, 16, 25, … __36_, __49_
perfect squares
5) 1, 3, 6, 10, 15, … __21_. _28__
add 2, then 3, then 4, etc.
6) 1, 3, 7, 15, 31, 63, … _127_, _255_
add 2, then 4, then 8, then 16, etc.
7) 1, 1, 2, 3, 5, 8, … __13_, _21__
add the preceding two terms
Fibonacci Sequence
8) 3, 5, 9, 15, 23, … _33__, _45__
add 2, then 4, then 6, then 8, etc.
Inductive reasoning can also be used to find missing terms in sequences and patterns dealing
with pictures. Draw the next two figures for each of the following and describe the pattern.
9)
10)
The student should draw a shaded triangle,
then an unshaded square.
The student should draw two
shaded pentagons.
11)
The student should draw a circle with an inscribed pentagon. The points on the circles increase
by one in each picture, which are connected to make polygons.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 2
Unit 1, Activity 3, Linear or Non-linear
‘Tis Linear or Not linear; That is the Question’
Directions: Decide whether each of the given rules, sequences, or tables represents a linear or
non-linear pattern. Place a check under the column which corresponds to your decision. Be
prepared to explain why you made your decision.
Is the given pattern
Linear
1)
2,5,8,11,14,...
2)
1, 2, 4,8,16,...
3)
3n 1
4)
x
1
2
3
4
5
y
18
15
12
9
?
5)
n2  1
6)
15, 10, 6, 3, 1,...
7)
8)
x
1
2
3
4
5
y
100
50
25
12.5
?
Non-linear
n4
2
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 3
Unit 1, Activity 3, Linear or Non-linear with Answers
‘Tis Linear or Not linear; That is the Question’
Directions: Decide whether each of the given rules, sequences, or tables represents a linear or
non-linear pattern. Place a check under the column which corresponds to your decision. Be
prepared to explain why you made your decision.
Is the given pattern
Linear
Non-linear

1)
2,5,8,11,14,...
2)
1, 2, 4,8,16,...
3)
3n 1
4)
x
1
2
3
4
5
y
18
15
12
9
?



5)
n2  1

6)
15, 10, 6, 3, 1,...

7)
8)
x
1
2
3
4
5
y
100
50
25
12.5
?
n4
2
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008


Page 4
Unit 1, Activity 3, Using Rules to Generate a Sequence
Linear versus Non-linear Relationships
Linear data are data that ____________________________
Consider a few different patterns.
1)
Term n
Value n-3
2)
3)
4)
5)
1
-2
2
-1
3
0
4
5
6
7
8
Term n
1
Value 2n+3 5
2
7
3
9
4
5
6
7
8
Term n
1
Value 3n+1 4
2
7
3
10
4
5
6
7
8
Term n
Value n2
1
1
2
4
3
9
4
5
6
7
8
Term n
Value n3
1
1
2
8
3
27
4
5
6
7
8
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 5
Unit 1, Activity 3, Using Rules to Generate a Sequence
Questions to answer:
6) Which patterns had common differences (the same number added over and over)? Does that
number appear in the rule?
7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern was
rewritten in this form, how should m be interpreted?
Graph each of the sequences above on a sheet of graph paper to determine if they are linear or
not linear.
8) Which sequences produced a line? What did these sequences have in common?
9) Which sequences did not produce a line? What did these sequences have in common?
10) Write a conjecture about all linear relationships and all non-linear relationships based on
your examples above.
Are the following sequences linear or non-linear?
11) -1.5, -1, -0.5, 0, 0.5, …
12) 4, 10, 18, 28, 40, …
13) 2, 1, 2/3, ½, 2/5, …
14) 1, 4, 7, 10, 13, …
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 6
Unit 1, Activity 3, Using Rules to Generate a Sequence with Answers
Linear versus Non-linear relationships
Linear data are data that _forms a line when graphed__
Consider a few different patterns.
1)
Term n
Value n-3
1
-2
2
-1
3
0
4
1
5
2
6
3
7
4
8
5
4
11
5
13
6
15
7
17
8
19
4
13
5
16
6
19
7
22
8
25
4
16
5
25
6
36
7
49
8
64
6
216
7
343
8
512
Difference between the terms is 1
2)
Term n
1
Value 2n+3 5
2
7
3
9
Difference between the terms is 2
3)
Term n
1
Value 3n+1 4
2
7
3
10
Difference between the terms is 3
4)
Term n
Value n2
1
1
2
4
3
9
There is no common difference between terms
5)
Term n
Value n3
1
1
2
8
3
27
4
64
5
125
There is no common difference between terms
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 7
Unit 1, Activity 3, Using Rules to Generate a Sequence with Answers
Questions to answer:
6) Which patterns had common differences (the same number added over and over)? Does that
number appear in the rule?
Patterns 1, 2, and 3 had common differences. These numbers are the coefficients of n.
7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern was
rewritten in this form, how should m be interpreted?
The m stands for slope. If I rewrote the rule in the slope-intercept form it would tell me
the slope of the line which is the rate of change—how much y changes when x changes.
Graph each of the sequences above on a sheet of graph paper to determine if they are linear or
not linear.
8) Which sequences produced a line? What did these sequences have in common?
Patterns 1, 2, and 3; each of these patterns had a common difference which is the
coefficient of n.
9) Which sequences did not produce a line? What did these sequences have in common?
Patterns 4 and 5; these patterns did not have a common difference.
10) Write a conjecture about all linear relationships and all non-linear relationships based on
your examples above.
Patterns which represent linear relationships will have a common difference between
terms. Patterns which are non-linear will not have a common difference between terms.
Are the following sequences linear or non-linear?
11) -1.5, -1, -0.5, 0, 0.5, …
12) 4, 10, 18, 28, 40, …
Linear
13) 2, 1, 2/3, ½, 2/5, …
Non-linear
14) 1, 4, 7, 10, 13, …
Non-linear
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Linear
Page 8
Unit 1, Activity 4, Generating the nth Term for Picture Patterns
Date ___________
Name ________________________
Directions: Find the indicated term for each of the patterns below.
1)
How many sides will the 15th term have?
2)
What will the 23rd figure look like?
3)
What is the 50th term of the sequence above?
4)
What is the 103rd term of the sequence?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 9
Unit 1, Activity 4, Generating the nth Term for Picture Patterns with Answers
Date ___________
Name ________________________
Directions: Find the indicated term for each of the patterns below.
1)
How many sides will the 15th term have?
Solution: n + 2; 17 sides Add two to the figure number, to determine the number of sides.
For example, the 3rd figure has 5 sides.
2)
What will the 23rd figure look like?
Solution: Since the pattern repeats after four figures, students should realize that every term
that is a multiple of four will look like the fourth figure. The nearest multiple to 23 is 20; the
students should then continue the pattern—it is the 3rd figure.
3)
What is the 50th term of the sequence above?
Solution: The shapes repeat after 3 terms so 48 is the closest multiple of 3 to 50, so the
shape is a square. The square is not shaded because the even terms are not shaded.
4)
What is the 103rd term of the sequence?
Solution: The pattern repeats after five terms. The 100th term is the fifth figure, so the 103rd
term is the third figure.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 10
Unit 1, Activity 5, Square Figurate Numbers
Date _____________
Name ________________________
Square Numbers
Consider the following sequence:
1) What is the number pattern?
2) Is it linear? Why?
3) What is the formula to find the nth term in this set? What would be the 25th term?
4) How does each number relate to the area of a square?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 11
Unit 1, Activity 5, Square Figurate Numbers with Answers
Date _____________
Name ________________________
Square Numbers
Consider the following sequence:
1) What is the number pattern?
1, 4, 9, 16, 25
2) Is it linear? Why?
It is not linear because the difference between consecutive terms is not constant.
3) What is the formula to find the nth term in this set? What would be the 25th term?
Formula: n 2 ; the 25th term is 625.
4)How does each number relate to the area of a square?
The area of a square is s 2 where s is the measure of the side. In each of the squares, the measure
of the sides are the same and they increase by one each time.
Therefore the area is 22, 32, 42, … n 2 .
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 12
Unit 1, Activity 5, Rectangular Figurate Numbers
Date _____________
Name ________________________
Rectangular Numbers
Consider the following:
1) What is the number pattern?
2) Is it linear? Why?
3) What is the formula to find the nth term in this set? What would be the 25th term?
4) How does each number relate to the area of a rectangle?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 13
Unit 1, Activity 5, Rectangular Figurate Numbers with Answers
Date _____________
Name ________________________
Rectangular Numbers
Consider the following:
1) What is the number pattern?
2, 6, 12, 20, 30
2) Is it linear? Why?
It is not linear because the difference between consecutive terms is not constant.
3) What is the formula to find the nth term in this set? What would be the 25th term?
Formula: n 2  n or n  n  1 ; the 25th term is 650.
4) How does each number relate to the area of a rectangle?
Each rectangle has a height the same as the figure number and a base which is one greater than
the height; therefore the number of dots needed for any figure is the same as the area of the
rectangle, n(n+1), where n is the height and the base is one more than the height.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 14
Unit 1, Activity 5, Triangular Figurate Numbers
Date _____________
Name ________________________
Triangular Numbers
Consider the following:
1) What is the number pattern?
2) Is it linear? Why?
3) What is the formula to find the nth term in this set? What would be the 25th term?
4) How does each number relate to the area of a triangle?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 15
Unit 1, Activity 5, Triangular Figurate Numbers with Answers
Date _____________
Name ________________________
Triangular Numbers
Consider the following:
1) What is the number pattern?
1, 3, 6, 10, 15
2) Is it linear? Why?
It is not linear because the difference between consecutive terms is not constant.
3) What is the formula to find the nth term in this set? What would be the 25th term?
Formula:
n  n  1
n2  n
or
or 0.5n2  0.5n ; the 25th term is 325.
2
2
4) How does each number relate to the area of a triangle?
1
The area of a triangle is half the area of a rectangle, A  bh , so if we take the formula for
2
rectangular numbers we can divide it by 2 to get the area of a triangle with the same base as its
corresponding rectangle.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 16
Unit 2, Activity 1, Logic Puzzle
Name: ____________________
Date: _________________
Directions: Using the given information try to solve the following puzzle.
Don, Frank, Jenny, and Ken each come from one state: Alaska, Maine,
Montana, or Oklahoma. Each speaks one primary language: English,
French, Russian, or Spanish. Each has one of four pets—a chinchilla, a
dog, a hamster, or a turtle. Use this information and the following clues to
determine where each person lives, what their primary language is, and
which pet they own.
1.
Frank needed a language book to write a letter to the Alaskan.
2.
The kid from Oklahoma has a mammal for her pet.
3.
The Alaskan found his pet outside his door in a snow bank.
4.
The French-speaking boy lives east of Oklahoma.
5.
The Russian-speaking boy wants to write the kid from Montana, but
he doesn’t speak his language.
6.
Don bought his pet in Peru.
7.
Ken does not own a hamster.
8.
The dog’s owner wrote a letter in Russian to the kid in Oklahoma, but
she couldn’t understand it.
9.
Don had to travel west to meet Jenny.
10.
Frank is learning Spanish at school.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 17
Unit 2, Activity 1, Logic Puzzle with Answers
Name: ____________________
Date: _________________
Directions: Using the given information try to solve the following puzzle.
Don, Frank, Jenny, and Ken each come from one state: Alaska, Maine,
Montana, or Oklahoma. Each speaks one primary language: English,
French, Russian, or Spanish. Each has one of four pets—a chinchilla, a
dog, a hamster, or a turtle. Use this information and the following clues to
determine where each person lives, what their primary language is, and
which pet they own.
1.
Frank needed a language book to write a letter to the Alaskan.
2.
The kid from Oklahoma has a mammal for her pet.
3.
The Alaskan found his pet outside his door in a snow bank.
4.
The French-speaking boy lives east of Oklahoma.
5.
The Russian-speaking boy wants to write the kid from Montana, but
he doesn’t speak his language.
6.
Don bought his pet in Peru.
7.
Ken does not own a hamster.
8.
The dog’s owner wrote a letter in Russian to the kid in Oklahoma, but
she couldn’t understand it.
9.
Don had to travel west to meet Jenny.
10.
Frank is learning Spanish at school.
Solution: Don lives in Maine, owns a chinchilla, and speaks French; Frank lives in Montana,
owns a turtle, and speaks English; Jenny lives in Oklahoma, owns a hamster, and speaks
Spanish; Ken lives in Alaska, owns a dog, and speaks Russian. These answers also apply to the
Logic Puzzle with Grid BLM.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 18
Unit 2, Activity 1, Logic Puzzle with Grid
Name: _____________________
Date: _________________
Directions: Using the given information try to solve the following puzzle using the grid below.
Don, Frank, Jenny, and Ken each come from one state: Alaska, Maine, Montana, or Oklahoma.
Each speaks one primary language: English, French, Russian, or Spanish. Each has one of four
pets—a chinchilla, a dog, a hamster, or a turtle. Use this information and the following clues to
determine where each person lives, what their primary language is, and which pet they own.
1.
Frank needed a language book to write a letter to the Alaskan.
2.
The kid from Oklahoma has a mammal for her pet.
3.
The Alaskan found his pet outside his door in a snow bank.
4.
The French-speaking boy lives east of Oklahoma.
5.
The Russian-speaking boy wants to write the kid from Montana, but he doesn’t speak his
language.
6.
Don bought his pet in Peru.
7.
Ken does not own a hamster.
8.
The dog’s owner wrote a letter in Russian to the kid in Oklahoma, but she couldn’t
understand it.
9.
Don had to travel west to meet Jenny.
10.
Frank is learning Spanish at school.
AK ME MT OK Eng. Fr. Russ. Sp. Chin. Dog Ham. Turtle
Don
Frank
Jenny
Ken
Chinchilla
Dog
Hamster
Turtle
English
French
Russian
Spanish
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 19
Unit 2, Activity 10, Proof Process Guide
Please print two copies of each proof—one to be cut up and one to be used as an
Answer Key.
Cut the statements and reasons in the following proofs into strips and put them in
envelopes to have the students arrange in the correct order. If students need help
identifying the strips as either a statement or reason, put all of the statements from
one proof in one envelope and the reasons for the proof in a separate envelope.
Label the envelopes Statements Proof # and Reasons Proof #. The statements and
reasons are not numbered below, but the order in which they are presented is the
order that the students should have when their work is completed.
Proof #1
Given: 4  x  2  52
Prove: x  15
Statements
Reasons
4  x  2  52
Given
4 x  8  52
Distributive Property
4 x  8  8  52  8
Addition Property
4 x  60
Substitution Property
4 x  60
4 4
Division Property
x  15
Substitution Property
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 20
Unit 2, Activity 10, Proof Process Guide
Proof #2
Given: 3 2  4a   3 a  6 123
Prove: x  9
Statements
Reasons
3 2  4a   3 a  6 123
Given
6 12a  3a 18  123
Distributive Property
12 15a  123
Substitution Property
12 15a 12  123 12
Addition Property
15a  135
Substitution Property
15a  135
15 15
Division Property
x  9
Substitution Property
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 21
Unit 2, Activity 10, Proof Process Guide
Proof #3
Given: 2 x  6  2
5
Prove: x  2
Statements
2x  6  2
5


5  2 x  6   5  2
 5 
Reasons
Given
Multiplication Property
2 x  6  10
Substitution Property
2 x  6  6  10  6
Subtraction Property
2x  4
Substitution Property
2x  4
2 2
Division Property
x2
Substitution Property
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 22
Unit 2, Activity 10, Proof Process Guide
Proof #4
Given: 4  1 a  7  a
2
2
Prove: a  1
Statements
4 1 a  7 a
2
2




2 4  1 a   2 7  a 
2  2 

Reasons
Given
Multiplication Property
8  a  7  2a
Substitution Property
8  a  8  7  2a  8
Subtraction Property
a  2a 1
Substitution Property
a  2a  2a 1 2a
Addition Property
a  1
Substitution Property
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 23
Unit 3, Activity 7, Parallel Line Facts
Date _________
Name ________________________
Use the given diagram to complete the steps and answer the questions.
1.
Draw a line through vertex B so that the line is parallel to AC .
2.
Given the diagram above with the parallel line drawn through B, prove that
mBAC  mABC  mACB  180 .
3.
Using the same diagram above, extend
AC
so that it is a line. Draw two points on
AC : one to the left of A labeled D and the other to the right of C and label it E. Using
the new diagram, prove that
mBAD  mABC  mACB .
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 24
Unit 3, Activity 7, Parallel Line Facts
4.
Remember, the area of a triangle can be written as A 
1
bh . Use the diagram below to
2
answer the following questions.
a. Using D, draw triangle ADC.
b. Choose a point anywhere on BD and label it E. Draw triangle AEC.
c. What do you notice about the base of each triangle:
ABC , AB ' C , ADC , and AEC ?
d. What do you notice about the height of each triangle?
e. What conjecture can you make about the area of any triangle that would be drawn
between these parallel lines if A and C are not moved to different positions? Explain your
reasoning.
f. Would your conjecture still be true if you were able to choose any three points on the
two lines to draw your triangles? Explain your reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 25
Unit 3, Activity 7, Parallel Line Facts with Answers
Date _________
Name ________________________
Use the given diagram to complete the steps and answer the questions.
1.
Draw a line through vertex B so that the line is parallel to AC . Locate one point to the
left of B and label it L. Locate one point to the right of B and label it R.
2.
Given the diagram above with the parallel line drawn through B, prove that
mBAC  mABC  mACB  180 .
Given that BR AC , we know that alternate interior angles are congruent. So,
BAC  ABL and ACB  CBR . By definition of congruence,
mBAC  mABL and mACB  mCBR . Because ABL, ABC , and CBR are
adjacent and form a line, mABL  mABC  mCBR  180 . Using the substitution
property of equality, we now have mBAC  mABC  mACB  180 .
3.
Using the same diagram above, extend AC so that it is a line. Draw two points on AC :
one to the left of A labeled D and the other to the right of C and label it E. Using the new
diagram, prove that mBAD  mABC  mACB .
Given that BR AC , we know that alternate interior angles are congruent. So,
BAD  ABR and ACB  CBR . By definition of congruence, we also know that
mBAD  mABR and mACB  mCBR . Using the Angle Addition Postulate, we
know mABR  mABC  mCBR . Next, using the Substitution Property of Equality,
we find mABR  mABC  mACB . Using the Substitution Property of Equality one
more time, we get mBAD  mABC  mACB .
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 26
Unit 3, Activity 7, Parallel Line Facts with Answers
4.
1
Remember, the area of a triangle can be written as A  bh . Use the diagram below to
2
answer the following questions.
a. Using D, draw triangle ADC.
b. Choose a point anywhere on BD and label it E. Draw triangle AEC.
c. What do you notice about the base of each triangle:
ABC , AB ' C , ADC , and AEC ?
The base of all four triangles is segment AC. The measure of the base doesn’t change.
d. What do you notice about the height of each triangle?
Since the distance between parallel lines is equal everywhere, the height of all four
triangles is the same.
e. What conjecture can you make about the area of any triangle that would be drawn
between these parallel lines if A and C are not moved to different positions? Explain your
reasoning.
Since both the base and height of these triangles are the same, they will have the same
area.
f. Would your conjecture still be true if you were able to choose any three points on the
two lines to draw your triangles? Explain your reasoning.
No the conjecture would not necessarily work. If the measure of the base were changed
each time the area of each triangle would also change despite the fact that the height
remained the same.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 27
Unit 3, General Assessment, Scrapbook Rubric
Parallel and Perpendicular Lines Scrapbook
4 points
3 points
2 points
1 point
0 points
Quantity (24)
Minimum of 3
photos per term
(Parallel or
Perpendicular)
Only two
photos/
pictures per
term
Only one
photo/picture
per term
Only one
picture to
demonstrate
both terms
No photos ______
or pictures x 6
Quality (24)
Photos are of
excellent
quality; clear;
description is
written clearly
Photos/
pictures are of
good quality;
description is
clear but
missing some
elements
Photos/pictures
are grainy;
term is not
clearly
depicted in
picture;
description is
vague
Photos/pictures No
______
do not depict
description x 6
term at all;
given
description
only gives
definition
Title Page (8)
Excellent
Typed; missing
quality; typed;
date or class
includes project period
title, date, and
class period
Handwritten
with all
information; or
typed and
missing date
and class
period
Handwritten
No title
and missing
page
date and class
period; missing
title (typed with
all other info)
______
x2
Reflection (12)
Typed; tells
what the student
learned from
project;
grammatically
correct
Typed; 1-2
grammar
errors; some
evidence of
learning
Handwritten; 34 grammar
errors; vague
evidence of
learning
Handwritten; 5- No
6 grammar
reflection
errors; little to
no evidence of
learning
______
x3
Neatness/
Creativity (8)
Typed; clean;
neatly bound
pages; original
project title;
attractive; etc.
Typed; project Handwritten;
name not
project is less
original; some than attractive;
pages loose
Dirty, crumpled Pages are ______
pages; if
not bound x 2
handwritten
there are
scratchouts or
places with
liquid paper
Timeliness (8)
Turned in on
time
Turned in one
day late
Turned in three Turned in ______
days late
more than x 2
three days
late
CATEGORY
Turned in two
days late
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Score
Page 28
Comments
Unit 3, Activity 2, Specific Assessment, What’s My Line?
What’s My Line?
On the attached page you have been given your own individual line and a point. Every line is
unique and has its own unique equation. It is your job to find out as much about your line as
possible. Listed below is the information that you must determine about your line.
1.
Locate, draw, and label an x-axis and y-axis.
2.
Find two points on your line. Label their (x,y) coordinates. Using the two points find the
slope of your line. Show your calculations below. Write the slope next to the line as
m=____.
3.
Determine the slope-intercept form of the equation for your line. Show your
calculations below. Label your line with the equation.
4.
Calculate the x and y-intercepts. Show your calculations below. On your graph,
identify and label by giving their (x,y) coordinates.
5.
Draw the line that is perpendicular to your line that passes through the point that was
given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this
equation. Show your calculations below.
6.
Draw the line that is parallel to your line that passes through the point that was given on
the page. Write the word parallel next to this line. Write the slope-intercept form of
the equation for the parallel line and label the line with this equation. Show your
calculations below.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 29
Unit 3, Activity 2, Specific Assessment, What’s My Line? with Answers
What’s My Line?
On the attached page you have been given your own individual line and a point. Every line is
unique and has its own unique equation. It is your job to find out as much about your line as
possible. Listed below is the information that you must determine about your line.
1.
Locate, draw, and label an x-axis and y-axis.
Will vary by student
2.
Find two points on your line. Label their (x,y) coordinates. Using the two points find the
slope of your line. Show your calculations below. Write the slope next to the line as
m=____.
2
7
2
Graph A: m  ; Graph B: m   ; Graph C: m  ;
2
3
5
5
3
Graph D: m  ; Graph E: m  
11
2
Other answers cannot be given since there are no axes on the graphs (students must draw
these in on their own wherever they choose). The placement of the axes determines other
answers.
Determine the slope-intercept form of the equation for your line. Show your
calculations below. Label your line with the equation.
Answers will depend on where x/y axes are drawn by each student.
3.
4.
Calculate the x and y-intercepts. Show your calculations below. On your graph,
identify and label by giving their (x,y) coordinates.
Answers will depend on where x/y axes are drawn by each student.
5.
Draw the line that is perpendicular to your line that passes through the point that was
given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this
equation. Show your calculations below.
3
2
5
Graph A: m   ; Graph B: m  ; Graph C: m   ;
2
2
7
2
11
Graph D: m   ; Graph E: m 
5
3
6.
Draw the line that is parallel to your line that passes through the point that was given on
the page. Write the word parallel next to this line. Write the slope-intercept form of
the equation for the parallel line and label the line with this equation. Show your
calculations below.
2
7
2
Graph A: m  ; Graph B: m   ; Graph C: m  ;
2
3
5
5
3
Graph D: m  ; Graph E: m  
11
2
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 30
Unit 3, Activity 2, Specific Assessment, What’s My Line? Graph A
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 31
Unit 3, Activity 2, Specific Assessment, What’s My Line? Graph B
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 32
Unit 3, Activity 2, Specific Assessment, What’s My Line? Graph C
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 33
Unit 3, Activity 2, Specific Assessment, What’s My Line? Graph D
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 34
Unit 3, Activity 2, Specific Assessment, What’s My Line? Graph E
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 35
Unit 3, Activity 2, Specific Assessment, What’s My Line? Rubric
What’s My Line?
Rubric
Points
Possible
Description
I.
x-axis and y-axis drawn and labeled
5
II.
Slope of the line
A. 2 points labeled with (x,y) coordinates
B. Slope calculated correctly
C. Labeled line with slope
2
6
1
Equation of the line
A. Calculated correctly
B. Labeled on graph
6
1
x and y-intercepts
A. Calculated correctly
B. Labeled on graph
6
2
Perpendicular Line
A. Line drawn correctly
B. Slope of line correctly identified
C. y-intercept calculated correctly
D. Equation written correctly based on calculations shown
C. Line labeled with perpendicular and equation
3
2
2
2
2
Parallel Line
A. Line drawn correctly
B. Slope of line correctly identified
C. y-intercept calculated correctly
D. Equation written correctly based on calculations shown
C. Line labeled with parallel and equation
3
2
2
2
2
Following directions
A. Cover sheet
B. Stapled
C. Submitted on or before due date
3
3
3
III.
IV.
V.
VI.
VII.
Total
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Points
Earned
60
Page 36
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the triangles congruent.
D
A
Given: X is the midpoint of BD
X is the midpoint of AC
X
Prove: DXC  BXA
C
B
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 37
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides congruent.
D
A
Given: X is the midpoint of BD
X is the midpoint of AC
X
Prove: DA  BC
C
B
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 38
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the angles congruent.
M
A
H
T
Given: MA TH ; HM TA
Prove: H  A
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 39
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides congruent.
M
A
H
T
Given: MA TH ; MA  TH
Prove: MH  AT
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 40
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the triangles congruent.
I
Given: X is the midpoint of KT
IE  KT
Prove:
KXE  TXE
K
X
T
E
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 41
Unit 4, Activity 6, Proving Triangles Congruent
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides congruent.
I
Given: X is the midpoint of KT
IE  KT
K
X
T
Prove: KI  TI
E
Write your proof below. Be sure to include all logical reasoning.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 42
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the triangles congruent.
D
A
Given: X is the midpoint of BD
X is the midpoint of AC
X
Prove: DXC  BXA
C
B
Write your proof below. Be sure to include all logical reasoning.
Statements
1. X is the midpoint of BD ; X is the midpoint
of AC .
2. XD  XB; XA  XC
3. DXC  BXA
4. DXC  BXA
Reasons
1. Given
2. Midpoint Theorem
3. Vertical angles are congruent.
4. SAS Postulate
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 43
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides triangles congruent.
D
A
Given: X is the midpoint of BD
X is the midpoint of AC
X
Prove: DA  BC
C
B
Write your proof below. Be sure to include all logical reasoning.
Statements
1. X is the midpoint of BD ; X is the midpoint
of AC .
2. XD  XB; XA  XC
3. DXA  BXC
4. DXA  BXC
5. DA  BC
Reasons
1. Given
2. Midpoint Theorem
3. Vertical angles are congruent.
4. SAS Postulate
5. CPCTC
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 44
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the angles triangles congruent.
M
A
H
T
Given: MA TH ; HM TA
Prove: H  A
Write your proof below. Be sure to include all logical reasoning.
1.
2.
3.
4.
5.
6.
Statements
MA TH ; HM TA
HMT  ATM
AMT  HTM
MT  MT
MHT  TAM
H  A
Reasons
1. Given
2. Alternate Interior Angles Theorem
3. Alternate Interior Angles Theorem
4. Reflexive Property of Congruence
5. ASA Postulate
6. CPCTC
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 45
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides triangles congruent.
M
A
H
T
Given: MA TH ; MA  TH
Prove: MH  AT
Write your proof below. Be sure to include all logical reasoning.
1.
2.
3.
4.
5.
Statements
MA TH ; MA  TH
MAH  THA
AH  AH
MHT  TAM
MH  TA
Reasons
1. Given
2. Alternate Interior Angles Theorem
3. Reflexive Property of Congruence
4. SAS Postulate
5. CPCTC
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 46
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the triangles congruent.
I
Given: X is the midpoint of KT
IE  KT
Prove:
K
KXE  TXE
X
T
E
Write your proof below. Be sure to include all logical reasoning.
Statements
1. X is the midpoint of KT ; IE  KT
2. KX  TX
3. mKXE  90; mTXE  90
4. mKXE  mTXE
5. KXE  TXE
6. EX  EX
7. KXE  TXE
Reasons
1. Given
2. Midpoint Theorem
3. Definition of perpendicular lines
4. Substitution Property of Equality
5. Definition of congruence
6. Reflexive Property of Congruence
7. SAS Postulate
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 47
Unit 4, Activity 6, Proving Triangles Congruent with Answers
Group Members ________________________________________
Date _____________
Proving Triangles Congruent
Using the given diagram and information, prove two of the sides triangles congruent.
I
Given: X is the midpoint of KT
IE  KT
K
X
T
Prove: KI  TI
E
Write your proof below. Be sure to include all logical reasoning.
Statements
1. X is the midpoint of KT ; IE  KT
2. KX  TX
3. mKXI  90; mTXI  90
4. mKXI  mTXI
5. KXI  TXI
6. IX  IX
7. KXI  TXI
8. KI  TI
Reasons
1. Given
2. Midpoint Theorem
3. Definition of perpendicular lines
4. Substitution Property of Equality
5. Definition of congruence
6. Reflexive Property of Congruence
7. SAS Postulate
8. CPCTC
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 48
Unit 4, Activity 11, Angle and Side Relationships
Group Members__________________
Date __________
Use the following charts to record the measurement data from the triangles.
Which group did the three triangles come from? _______________________________________
Triangle # 1
Name: _____________________
Angle Name
Angle Measure
1.
2.
3.
Side Name
Side Measure
1.
2.
3.
List the names of the angles and sides in order from largest to smallest on the lines below:
Angles
_________
_________
_________
Sides
_________
_________
_________
Triangle #2
Name: _____________________
Angle Name
Angle Measure
1.
2.
3.
Side Name
Side Measure
1.
2.
3.
List the names of the angles and sides in order from largest to smallest on the lines below:
Angles
_________
_________
_________
Sides
_________
_________
_________
Triangle #3
Name: _____________________
Angle Name
1.
2.
3.
Angle Measure
Side Name
Side Measure
1.
2.
3.
List the names of the angles and sides in order from largest to smallest on the lines below:
Angles
_________
_________
_________
Sides
_________
_________
_________
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 49
Unit 4, Activity 11, Angle and Side Relationships
Look at the measures of the angles and find the largest angle. Locate the side opposite the largest
angle. How does the measure of the side opposite the largest angle compare to the measures of
the other two sides?
Look at the measures of the sides and find the shortest side. Locate the angle opposite of the
shortest side. How does the measure of the angle opposite the shortest side compare to the
measures of the other two angles?
What conjecture can you draw from these observations?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 50
Unit 4, Activity 15, Quadrilateral Process Guide
Date______________
Name___________________
Partner’s Name___________________
Use the following guide to investigate the relationships that occur in different convex
quadrilaterals.
1.
Which quadrilateral are you working with?
2.
Measure all four angles and all four sides of the given quadrilateral and record the
information below.
Angle Measures
3.
Side Measures
Resize the quadrilateral by dragging the vertices. Measure the angles and sides again and
record the information.
Angle Measures
Side Measures
4.
Continue to resize the quadrilateral and make measurements for this quadrilateral. After
creating a minimum of 5 different sized quadrilaterals, make conjectures about the
measures of the sides and angles of any quadrilateral of this type. You should have
multiple conjectures.
5.
Construct the diagonals of the quadrilateral and answer the following questions:
a.) Do the diagonals bisect each other?
b.) Are the diagonals congruent?
c.) Are the diagonals perpendicular?
d.) Do the diagonals bisect the angles of the quadrilateral?
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 51
Unit 4, Activity 17, Quadrilateral Family
Date____________
Name___________________
Fill in the names of the quadrilaterals so that each of the following is used exactly once.
Parallelogram
Square
Trapezoid
Isosceles Trapezoid
Kite
Quadrilateral
Rectangle
Rhombus
If you follow the arrows from top to bottom, the properties of each figure are also properties of
the figure that follows it. For example, the properties of a parallelogram are also properties of a
rectangle.
If you reverse the arrows from bottom to top, every figure is also the one that precedes it. For
example, a square is also a rhombus and a rectangle since it is connected to them both.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 52
Unit 4, Activity 17, Quadrilateral Family with Answers
Date____________
Name___________________
Fill in the names of the quadrilaterals so that each of the following is used exactly once.
Parallelogram
Square
Trapezoid
Isosceles Trapezoid
Kite
Quadrilateral
Rectangle
Rhombus
Quadrilateral
Trapezoid
Kite
Isosceles
Trapezoid
Parallelogram
Rectangle
Rhombus
Square
If you follow the arrows from top to bottom, the properties of each figure are also properties of
the figure that follows it. For example, the properties of a parallelogram are also properties of a
rectangle.
If you reverse the arrows from bottom to top, every figure is also the one that precedes it. For
example, a square is also a rhombus and a rectangle since it is connected to them both.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 53
Unit 4, Activity 6, Specific Assessment
Instructions for Product Assessment
Activity 6
Your task is to design a tile in the 5-inch by 5-inch squares provided on the next two pages.
There are two parts to this project.
Part I:
The drawings on your tile must meet certain specifications. You must have the following, and
you will be graded on the accuracy of the following.
1.) 2 congruent obtuse triangles which demonstrate congruency by ASA
2.) 2 congruent scalene triangles which demonstrate congruency by SSS
3.) 2 congruent isosceles right triangles which demonstrate congruency by SAS
4.) 2 congruent acute triangles which demonstrate congruency by AAS
5.) 1 equilateral triangle
You should have a minimum of 9 triangles in your design (i. e. your two acute triangles
CANNOT double as your two scalene triangles). You may add other shapes once you are sure
you have the required 9 triangles above.
On Part I, you must label and mark each pair of triangles according to one of the methods
indicated in the directions above (see example below).
2 scalene triangles congruent by SSS
D
AB  DB
B
AC  DC
BC  BC
A
ABC  DBC
C
Part II
For Part II, you are to redraw your tile (without the markings and labels) in the square on the
second page and COLOR it. Cut the tile out of the page and put your name, number and hour ON
THE BACK! Do NOT glue it to another page, and do NOT staple it to part one. If you do not
complete part two, the entire project will be returned to you, and you will lose one letter grade
for each day late!!
DUE DATE:
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 54
Unit 4, Activity 6, Specific Assessment
Part I
Obtuse Triangles (ASA)
Scalene Triangles (SSS)








Acute Triangles (AAS)
Equilateral Triangle
Isosceles Right Triangles
(SAS)








Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 55
Unit 4, Activity 6, Specific Assessment
Part II
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 56
Unit 4, Activity 6, Specific Assessment Rubric
Activity 6 Product Assessment Rubric
This is a checklist for evaluating your tile design. Your grade will be a percentage based on the
number of requirements met.
Are the following present?
1.) 2 congruent obtuse triangles
2.) 2 congruent scalene triangles
3.) 2 congruent isosceles right triangles
4.) 2 congruent acute triangles
5.) 1 equilateral triangle
[
[
[
[
[
] yes
] yes
] yes
] yes
] yes
40%
[ ] no
[ ] no
[ ] no
[ ] no
[ ] no
Are the triangles marked by the correct method?
6.) ALL triangles labeled
7.) Obtuse congruent by ASA
8.) Scalene congruent by SSS
9.) Isosceles right congruent by SAS
10.) Acute congruent by AAS
[
[
[
[
[
] yes
] yes
] yes
] yes
] yes
40%
[ ] no
[ ] no
[ ] no
[ ] no
[ ] no
[
[
[
[
[
20%
] yes [ ] no
] yes [ ] no
] yes [ ] no
] yes [ ] no
] yes [ ] no
Are the congruent triangles and parts
listed correctly (based on markings)?
11.) Obtuse triangles and parts
12.) Scalene triangles and parts
13.) Isosceles right triangles and parts
14.) Acute triangles and parts
15.) Equilateral triangle
Following directions and promptness (for each “no” below you will lose one percentage
point):
16.) Tile drawn on handout and name on handout
[ ] yes [ ] no
17.) Part two is colored
[ ] yes [ ] no
18.) Part two is cut out and NOT attached by
staple or glue
[ ] yes [ ] no
19.) Name, number, and hour on back of part 2
[ ] yes [ ] no
20.) Rubric turned in
[ ] yes [ ] no
21.) Turned in on time
[ ] yes [ ] no
Score [4(
) + 4(
) + 2(
)]/10 =
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 57
Unit 4, Activity 15, Specific Assessment
Venn Diagram for Assessment for Activity 14
Directions: Label the Venn diagram below with the name and the number representing the
properties for parallelograms. Remember, in a Venn Diagram each property should only be listed
once.
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
Diagonals are perpendicular.
All four angles are right angles.
Opposite angles are congruent.
Diagonals bisect a pair of opposite angles.
Diagonals are congruent.
Opposite sides are congruent.
Diagonals bisect each other.
All four sides are congruent.
Opposite sides are parallel.
Consecutive angles are supplementary.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 58
Unit 4, Activity 15, Specific Assessment with Answers
Answer Key
Parallelograms – 3, 6, 7, 9
Rhombii – 1, 4, 8
Squares
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Rectangles – 2, 5
Page 59
Unit 5, Activity 1, Striking Similarity
Using the grid provided below, transfer the polygons to the blank grid you were given. You may
use a straight edge to help you draw the sides.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 60
Unit 5, Activity 2, Similarity and Ratios
Name ____________________
Date ____________________
Follow the given directions to explore the relationships between side lengths, area, and volume
of similar figures.
1.)
Given an equilateral triangle, use pattern blocks to create a similar triangle so the ratio of
side lengths is 2:1.
a.) What is the ratio of areas of the two similar triangles?
b.) Using pattern blocks create a triangle similar to the original triangle so the ratio of
side lengths is 3:1. What is the ratio of the areas of these two similar triangles?
2.)
Use other pattern block shapes to create and investigate other similar polygons in the
same manner as described above, and record your findings in the table below.
description of similar shapes
.
.
.
.
ratio of sides
.
.
.
.
ratio of areas
.
.
.
.
3.)
Based on your investigations in the two activities, make a generalization. If the ratio of
sides of two similar polygons is n:1, what would the ratio of areas be?
4.)
Given a cube, create a similar cube with ratio of edges 2:1 using cm or sugar cubes. What
is the ratio of volumes? Create a similar cube with ratio of edges 3:1. What is the ratio of
volumes? If the edges of two cubes were in a ratio of n:1, what would the ratio of
volumes be? Record your findings in a table like the one below.
description of similar 3-D shapes
.
.
.
.
ratio of edges
.
.
.
.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
ratio of volumes
.
.
.
.
Page 61
Unit 5, Activity 2, Similarity and Ratios with Answers
Name ____________________
Date ____________________
Follow the given directions to explore the relationships between side lengths, area, and volume
of similar figures.
1.)
Given an equilateral triangle, use pattern blocks to create a similar triangle so the ratio of
side lengths is 2:1.
a.) What is the ratio of areas of the two similar triangles?
The ratio of the areas is 4:1.
b.) Using pattern blocks create a triangle similar to the original triangle so the ratio of
side lengths is 3:1. What is the ratio of the areas of these two similar triangles?
The ratio of the areas is 9:1.
2.)
Use other pattern block shapes to investigate other similar polygons in the same manner
as described above, and record your findings in the table below.
description of similar shapes
. Answers will vary
.
.
.
ratio of sides
.
.
.
.
ratio of areas
.
.
.
.
3.)
Based on your investigations in the two activities, make a generalization. If the ratio of
sides of two similar polygons is n:1, what would the ratio of areas be?
The ratios of the areas will be n2:1.
4.)
Given a cube, create a similar cube with ratio of edges 2:1 using cm or sugar cubes. What
is the ratio of volumes? Create a similar cube with ratio of edges 3:1. What is the ratio of
volumes? If the edges of two cubes were in a ratio of n:1, what would the ratio of
volumes be? Record your findings in a table like the one below.
description of similar 3-D shapes
. cube with face of 4 square units
. cube with face of 9 square units
. cube with face of n2 units
.
ratio of edges
2:1
3:1
n:1
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
ratio of volumes
8:1
27:1
n3:1
.
Page 62
Unit 5, Activity 4, Spotlight on Similarity
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 63
Unit 5, Activity 5, Specific Assessment, Making a Hypsometer
Format:
Individual or Small Group
Objectives:
Participants use the hypsometer and their knowledge of the proportional
relationship between similar triangles to determine the height of an object not
readily measured directly.
Materials:
For each hypsometer you need a straw, decimal graph paper, cardboard,
thread, a small weight, tape, a hole punch, scissors, and a meter stick.
Time Required: Approximately 90 minutes
Directions:
To make the hypsometer:
1) Tape a sheet of decimal graph paper to a piece of cardboard.
2) Tape the straw to the cardboard so that it is parallel to the top of the graph
paper.
3) Punch a hole in the upper right corner of the grid. Pass one end of the
thread through the hole and tape it to the back of the cardboard. Tie the
weight to the other end of the thread.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 64
Unit 5, Activity 5, Specific Assessment, Making a Hypsometer
To use the hypsometer:
4) Have a friend use a meter stick to measure the height of your eye from the
ground and the distance from you to the object to be measured.
5) Look through the straw at the top of the object you wish to measure. Your
friend should record the hypsometer reading as you remain steady and
continue to look through the straw at the top of the object
6) To find the height of the flagpole, recognize that triangles ABC and DEF
are similar. Thus, BC can be found using the following ratio —
AC BC

DF EF
Reference:
NCTM Addenda Series, Measurement in the Middle Grades
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 65
Unit 7, Activity 1, Vocabulary Self-Awareness
Word/Phrase
+

–
Definition/Formula
Example
radius
circumference
chord
area of a circle
central angle
arc
arc measure
arc length
major arc
minor arc
semicircle
sector
area of a sector
tangent
secant
sphere
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 66
Unit 7, Activity 1, Vocabulary Self-Awareness
surface area of a sphere
volume of a sphere
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 67
Unit 7, Activity 1, Vocabulary Self-Awareness with Answers
Word/Phrase
+

–
Definition/Formula
radius
a segment with one
endpoint at the center
of the circle and the
other endpoint on the
circle; one-half the
diameter
circumference
the distance around the
circle
chord
a segment whose
endpoints lie on the
circle
area of a circle
A   r2
central angle
an angle formed at the
center of a circle by
two radii
arc
a segment of a circle
arc measure
arc length
major arc
minor arc
Example
equal to the degree
measure of the central
angle;
arc measure
arc length

360
circumference
the distance along the
curved line making up
the arc;
arc measure
arc length

360
circumference
the longest arc
connecting two points
on a circle; an arc
having a measure
greater than 180
degrees
the shortest arc
connecting two points
on a circle; an arc
having a measure less
than 180 degrees
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 68
Unit 7, Activity 1, Vocabulary Self-Awareness with Answers
semicircle
sector
area of a sector
tangent
secant
sphere
an arc having a
measure of 180 degrees
and a length of onehalf of the
circumference; the
diameter of a circle
creates two semicircles
a plane figure bounded
by two radii and the
included arc of the
circle
N
A
 r 2 where N
360
is the measure of the
central angle
a line or segment which
intersects the circle at
exactly one point
a line or segment which
intersects the circle at
exactly two points
the locus of all points,
in space, that are a
given distance from a
given point called the
center
surface area of a sphere
SA  4 r 2
volume of a sphere
4
V   r3
3
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 69
Unit 7, Activity 4, Sample Notes
Date:
Period:
Topic: Circles
Parts of a circle:
radius
--one-half the diameter
--one endpoint is the center of the circle, the other is on the
circle
--used when finding the area of a circle
chord
--a segment whose endpoints are on the circle
diameter
--a chord which passes through the center of the circle
Formulas:
area of a circle
-- A   r 2
--r is the measure of the radius of the circle
circumference
-- C  2 r or C   d
--r is the measure of the radius and d is the measure of the
diameter
--these formulas are the same because d  2r .
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 70
Unit 7, Activity 4, Split-Page Notes Model
Date:
Period:
Topic: Central Angles and Arcs
central angle
--an angle whose vertex is the center of the circle and sides
are two radii
--the sum of all central angles in a circle is 360°
arc
--a segment of a circle
--created by a central angle or an inscribed angle
--has a degree measure (called arc measure)
--has a linear measure (called arc length)
minor arc
--an arc whose measure is less than 180 degrees
major arc
--an arc whose measure is greater than 180 degrees
semicircle
--an arc whose measure is exactly 180 degrees
--created by the diameter of the circle
--the arc length is one-half of the circumference of the circle
arc length
--the distance along the curved line making up the arc
arc measure
arc length
--found using the formula
.

360
circumference
--the formula can also be used to find the arc measure if the
arc length is known
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 71
Unit 7, Activity 5, Concentric Circles
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 72
Unit 7, Activity 8, Diameters and Chords
Date______________
Team Members___________________
Use the following guide to investigate the relationships that occur between the diameter
and chords of circles.
Investigation 1
1.
Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to
locate the center of the circle. Label the center C.
2.
Pick any two points on the circle (do NOT use the endpoints of the diameters). Label the
points G and H. Using a straightedge, draw the segment connecting G and H. What is
GH ? ________________________
3.
Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.
Unfold the paper and label the endpoints of the diameter just created as J and K.
4.
Draw CG and CH . Find the measure of GH . ________________________
5.
GH should have been divided into two smaller arcs—either GK and HK or
GJ and HJ . Find the measure of these two smaller arcs created by JK .
________________________________________________
6.
What is true about the two arcs measured in number five?
________________________________________________________________________
7.
Using a ruler, measure the radii CG and CH . What is the arc length of GH ?
________________________
What are the arc lengths of the two arcs measured in number five?
_______________________________________________________________________
What is true about the lengths of the two smaller arcs compared to the larger arc?
____________________________________________________________________
8.
Using a ruler, measure GH and the two smaller segments created by the intersection of
the diameter and the chord. ______________________________________________
9.
What conjecture can be made if the diameter of a circle is perpendicular to a chord?
________________________________________________________________________
Does this conjecture apply to the radii of a circle? Explain.
________________________________________________________________________
________________________________________________________________________
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 73
Unit 7, Activity 8, Diameters and Chords
Investigation 2
Follow the steps below in order to answer the questions that follow.
Step 1.
Use a compass to draw a large circle on patty paper. Cut out the circle.
Step 2.
Fold the circle in half.
Step 3.
Without opening the circle, fold the edge of the circle so it does not intersect the
first fold.
Step 4.
Unfold the circle and label the circle. Find the center by locating the point where
the compass was placed and label the center M. Darken the diameter which should
pass through the center. Locate the two other folds and darken the chords created
by these folds. Label one chord as GE and the other chord as TR .
Step 5.
Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold
again to bisect TR (lay point T onto R). Two diameters should have been formed.
Label the intersection point on GE as O and the intersection point on TR as Y.
Answer the following about Investigation 2.
1.
What is the relationship between MO and GE ? What is the relationship between
MY and TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this).
2.
Use a centimeter ruler to measure GE , TR, MO, and MY . What observation can be made?
3.
Make a conjecture about the distance that two chords are from the center when the chords
are congruent.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 74
Unit 7, Activity 8 , Diameter and Chords with Answers
Date______________
Team Members___________________
Use the following guide to investigate the relationships that occur between the diameter
and chords of circles.
Investigation 1
1.
Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to
locate the center of the circle. Label the center C.
2.
Pick any two points on the circle (do NOT use the endpoints of the diameters). Label the
points G and H. Using a straightedge, draw the segment connecting G and H. What is
GH ? A chord.
3.
Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.
Unfold the paper and label the endpoints of the diameter just created as J and K.
4.
Draw CG and CH . Find the measure of GH . Answers will vary.
5.
GH should have been divided into two smaller arcs—either GK and HK or
GJ and HJ . Find the measure of these two smaller arcs created by JK . Answers will
vary.
6.
What is true about the two arcs measured in number five? They have the same measure,
which means they are congruent.
7.
Using a ruler, measure the radii CG and CH . What is the arc length of GH ? Answers
will vary.
What are the arc lengths of the two arcs measured in number five? Answers will vary.
What is true about the lengths of the two smaller arcs compared to the larger arc? They
have the same measure, which means they are congruent.
8.
Using a ruler, measure GH and the two smaller segments created by the intersection of
the diameter and the chord. Answers will vary.
9.
What conjecture can be made if the diameter of a circle is perpendicular to a chord?
If the diameter of a circle is perpendicular to a chord, the diameter bisects the chord and
the arc.
Does this conjecture apply to the radii of a circle? Explain.
Yes, this conjecture also applies to the radii of a circle. A radius is a part of the diameter;
therefore, these properties are true for the radii.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 75
Unit 7, Activity 8 , Diameter and Chords with Answers
Investigation 2
Follow the steps below in order to answer the questions that follow.
Step 1.
Use a compass to draw a large circle on patty paper. Cut out the circle.
Step 2.
Fold the circle in half.
Step 3.
Without opening the circle, fold the edge of the circle so it does not intersect the
first fold.
Step 4.
Unfold the circle and label the circle. Find the center by locating the point where
the compass was placed and label the center M. Darken the diameter which should
pass through the center. Locate the two other folds and darken the chords created
by these folds. Label one chord as GE and the other chord as TR .
Step 5.
Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold
again to bisect TR (lay point T onto R). Two diameters should have been formed.
Label the intersection point on GE as O and the intersection point on TR as Y.
Answer the following about Investigation 2.
1.
What is the relationship between MO and GE ? What is the relationship between
MY and TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this).
MO and MY are perpendicular bisectors of GE and TR , respectively.
2.
Use a centimeter ruler to measure GE , TR, MO, and MY . What observation can be made?
GE  TR and MO  MY
3.
Make a conjecture about the distance that two chords are from the center when the chords
are congruent.
When two chords are congruent, they are equidistant from the center of the circle.
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 76
Unit 7, Activity 11, Tangents and Secants
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 77
Unit 7, Activity 11, Tangents and Secants
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 78
Unit 7, Activity 11, Tangents and Secants with Answers
1
mDEB
2
1
mCDB  mDB
2
mADB 






1
m AD  mCB
2
1
mAEC  m AC  mDB
2
mAED 
mE 
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
1
mBC  m AD
2
Page 79
Unit 7, Activity 11, Tangents and Secants with Answers
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
mE 
1
(mDB  m AD )
2
mA 
1
mBDC  mBC
2


Page 80
Unit 7, Activity 13, Surface Area of a Sphere
Blackline Masters, Geometry
Louisiana Comprehensive Curriculum, Revised 2008
Page 81