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1.1 Patterns and
Inductive
Reasoning
Geometry
Objectives/Assignment:

Find and describe patterns.

Use inductive reasoning to make real-life conjectures.
Finding & Describing Patterns

Geometry, like much of mathematics
and science, developed when people
began recognizing and describing
patterns. In this course, you will study
many amazing patterns that were
discovered by people throughout
history and all around the world. You
will also learn how to recognize and
describe patterns of your own.
Sometimes, patterns allow you to
make accurate predictions.
Ex. 1: Describing a Visual
Pattern

1
Describe the next figure in the pattern.
2
3
4
5
Ex. 1: Describing a Visual
Pattern - Solution

The sixth figure in the pattern has 6 squares in the
bottom row.
5
6
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence of numbers.
Predict the next number.
a.
1, 4, 16, 64
Many times in number pattern, it is easiest listing the
numbers vertically rather than horizontally.
Ex. 1: Describing a Number
Pattern

a.
Describe a
pattern in the
sequence of
numbers.
Predict the
next number.
1
4
16
64

How do you get to
the next number?

That’s right. Each
number is 4 times the
previous number. So,
the next number is

256, right!!!
Ex. 1: Describing a Number
Pattern
 How do you get to
the next number?

b.
Describe a
pattern in the
sequence of
numbers.
Predict the
next number.
-5
-2
4
13

That’s right. You add
3 to get to the next
number, then 6, then
9. To find the fifth
number, you add
another multiple of 3
which is +12 or

25, That’s right!!!
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence of
numbers. Predict the next number.

5,10,20,40…

80
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence of
numbers. Predict the next number.

1, -1, 2, -2, 3…

-3
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence of
numbers. Predict the next number.

15,12,9,6,…

3
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence of
numbers. Predict the next number.

1,2,6,24,120…

720
Ex. 1: Describing a Number
Pattern

Describe a pattern in the sequence.
Predict the next letter.

O,T,T,F,F,S,S,E,…

1,2,3,4,5,6,7,8,
N
9

Goal 2: Using Inductive
Reasoning

Much of the reasoning you need in geometry consists
of 3 stages:
1.
Look for a Pattern: Look at several examples. Use
diagrams and tables to help discover a pattern.
2.
Make a Conjecture. Use the example to make a
general conjecture. Okay, what is that?

3.
A conjecture is an unproven statement that is based on
observations. Discuss the conjecture with others.
Modify the conjecture, if necessary.
Verify the conjecture. Use logical reasoning to verify
the conjecture is true IN ALL CASES. (You will do this
in Chapter 2 and throughout the book).
Ex. 2: Making a Conjecture

Complete the conjecture.
Conjecture: The sum of the first n odd positive integers is
________?_________.
How to proceed:
List some specific examples and look for a pattern.
Ex. 2: Making a Conjecture
First odd positive integer:
1 = 1 =12
1 + 3 = 4 =22
1 + 3 + 5 = 9 =32
1 + 3 + 5 + 7 = 16 =42
The sum of the first n odd positive integers is n2.
Ex. 2: Making a Conjecture
First odd positive integer:
2 = 2 = 1x2
2 + 4 = 6 = 2x3
2 + 4 + 6= 12 = 3x4
2 + 4 + 6 + 8 = 20 = 4x5
2 + 4 + 6 + 8 + 10 = 30 = 5x6
The sum of the first 6 odd positive integers is what?

6x7 = 42
Note:

To prove that a conjecture is true, you need to prove it
is true in all cases. To prove that a conjecture is false,
you need to provide a single counter example. A
counterexample is an example that shows a conjecture
is false.
Ex. 3: Finding a
counterexample

Show the conjecture is false by finding a
counterexample.
Conjecture: For all real numbers x, the expressions x2 is
greater than or equal to x.
Ex. 3: Finding a
counterexample- Solution
Conjecture: For all real numbers x,
the expressions x2 is greater than or
equal to x.

The conjecture is false. Here is a
counterexample: (0.5)2 = 0.25, and
0.25 is NOT greater than or equal to
0.5. In fact, any number between 0
and 1 is a counterexample.
Ex. 3: Finding a counterexample
Conjecture: The sum of two numbers
is greater than either number.

The conjecture is false. Here is a
counterexample: -2 + -5 = -7

-7 is NOT greater than -2 or -5. In
fact, the sum of any two negative
numbers is a counterexample.
Ex. 3: Finding a counterexample
Conjecture: The difference of two
integers is less than either integer.

The conjecture is false. Here is a
counterexample: -2 - -5 = 3

3 is NOT less than -2 or -5. In fact,
the difference of any two negative
numbers is a counterexample.
Ex. 4: Using Inductive
Reasoning in Real-Life

Moon cycles. A full moon occurs when the moon is on
the opposite side of Earth from the sun. During a full
moon, the moon appears as a complete circle.
Ex. 4: Using Inductive
Reasoning in Real-Life

Use inductive reasoning and the information below to
make a conjecture about how often a full moon occurs.

Specific cases: In 2005, the first six full moons occur on
January 25, February 24, March 25, April 24, May 23 and
June 22.
Ex. 4: Using Inductive
Reasoning in Real-Life - Solution
A full moon occurs every 29 or 30 days.
 This conjecture is true. The moon
revolves around the Earth approximately
every 29.5 days.

Ex. 4: Using Inductive
Reasoning in Real-Life - NOTE

Inductive reasoning is very important to the study of
mathematics. You look for a pattern in specific cases
and then you write a conjecture that you think
describes the general case. Remember, though, that
just because something is true for several specific cases
does not prove that it is true in general.
Ex. 4: Using Inductive
Reasoning in Real-Life

Use inductive reasoning and the information below to
make a conjecture about the temperature.

The speed with which a cricket chirps is affected by the
temperature. If a cricket chirps 20 times in 14 seconds,
what is the temperature?

5 chirps
45 degrees
10 chirps
55 degrees
15 chirps
65 degrees
The temperature would be 75 degrees.
Find a pattern for each sequence.
Use the pattern to show the next
two terms or figures.
Use the table and inductive reasoning.
1. 3, –6, 18, –72, 360
–2160; 15,120
2.
3. Find the sum of the first 10 counting numbers.
55
4. Find the sum of the first 1000
counting numbers.
500,500
Show that the conjecture is false by finding one
counterexample.
5. The sum of two prime numbers is an
even number.
Sample: 2 + 3 = 5, and 5 is not even
-1
1.2 Drawing, Nets
& Other Models
Geometry
Objectives/Assignment:

Make isometric and orthographic drawings

Use nets and cross sections to analyze 3-dimensional
figures. (G.GMD.4)
GEOMETRY LESSON 1-2
Draw the next figure in each sequence.
1.
2.
GEOMETRY LESSON 1-2
Solutions
1. The block is rotating counterclockwise about its base.
The next figure is:
2. The block is rotating clockwise about its front face.
The next figure is:
There are many ways to represent a three
dimensional object. An isometric drawing is a way
to show three sides of a figure from a corner view.
You can use isometric dot paper to make an isometric
drawing. This paper has diagonal rows of dots that
are equally spaced in a repeating triangular pattern.
GEOMETRY LESSON 1-2
Make an isometric drawing of the cube structure below.
You will draw all the edges of the figure that you can see.
Start by drawing the front face of the figure. Next, draw the back
edges of the figure. Finally, fill in the right face, top faces, and left
edges.
Step 1:
Draw the front.
Step 2:
Draw the back.
Step 3:
Complete.
An orthographic drawing shows six different views
of an object: top, bottom, front, back, left side, and
right side. (In your book, we only use the top, front,
and right side views for an orthographic drawing.)
below.
Make an orthographic drawing of the isometric drawing
Orthographic drawings flatten the depth of a figure. An orthographic
drawing shows three views. Because no edge of the isometric drawing is
hidden in the top, front, and right views, all lines are solid.
A foundation drawing shows the base of a structure
and the height of each part.
A foundation drawing of the Sears Tower is shown below. The
building is made up of nine sections. The numbers tell how
many stories tall each section is.
49
89
65
109
109
89
65
89
49
Create a foundation drawing for the isometric drawing below.
To make a foundation drawing, use the top view of the orthographic
drawing.
(continued)
Because the top view is formed from 3 squares, show 3 squares in
the foundation drawing.
Identify the square that represents the tallest part. Write the number 2
in the back square to indicate that the back section is 2 cubes high.
Write the number 1 in each of the two front squares to indicate that
each front section is 1 cube high.
A net is a diagram of the surfaces of a
three-dimensional figure (a twodimensional pattern) that can be folded to
form the three-dimensional figure. To
identify a three-dimensional figure from a
net, look at the number of faces and the
shape of each face.
GEOMETRY LESSON 1-2
Is the pattern a net for a cube? If so, name two letters that
will be on opposite faces.
The pattern is a net because you can fold it to form a cube. Fold squares A
and C up to form the back and front of the cube. Fold D up to form a side.
Fold E over to form the top. Fold F down to form another side.
After the net is folded to form a cube, the following pairs of letters are on
opposite faces:
A and C are the back and front faces.
B and E are the bottom and top faces.
D and F are the right and left side faces.
GEOMETRY LESSON 1-2
Draw a net for the figure with a square base and four
isosceles triangle faces. Label the net with its dimensions.
Think of the sides of the square base as
hinges, and “unfold” the figure at these
edges to form a net. The base of each of
the four isosceles triangle faces is a side
of the square.
Use the figure at the right for Exercises 1–2.
1. Make an isometric drawing of the cube structure.
2. Make an orthographic drawing.
Drawings, Nets, and Other Models
3. Is the pattern a net for a cube? If so, name two letters that will be on opposite
faces.
yes; A and C, B and D, E and F
4. Draw a net for the figure.
Sample:
ASSIGNMENT:
Textbook:
1.1
page 6 #2-12 even
page 6 # 20-22 even
page 7 # 26-30 even
1.2
page 13 # 2-16 even
Page 716 (#1-17)