Transcript Teaching number systems - from workshop 10

Session1
Cultivating Skills for problem solving
Teaching the concept and notation of
Number Systems
using an understanding of basic rules
and skills approach.
Junior Certificate-All Levels
Leaving Certificate- Foundation Level
Leaving Certificate- Ordinary & Higher Level
Section 1
Number Systems
Prior Knowledge
Within Curriculum
Future
Strands
• Number
Systems (ℕ, ℤ & ℚ)
Subjects
Across
Real
Past
Strands
World
Assessment
Quiz
Time…………
The
Natural
numbers are…..
A.
The set of all whole numbers , positive, negative and 0.
B.
The set of all positive whole numbers (excluding 0).
C.
The set of all positive whole numbers (including 0).
The Integers are……
A.
The set of all whole numbers , positive, negative and 0.
B.
The set of all positive whole numbers only.
C.
The set of all negative whole numbers only.
Answer True or False to the following:
‘The natural numbers are a subset of the integers’.
TRUE
FALSE
Which number is not an integer?
A.
-1
B.
0
C.
𝟕
𝟏
D.
4.𝟑
The Rational numbers are…….
𝒑
𝒒
A.
Any number of the form , where p, q ∈ ℤ and q≠0.
𝒒
B.
Any number of the form
𝒑
,
𝒒
Any number of the form
𝒑
, where
𝒒
C.
where p, q ∈ ℤ .
p, q ∈ ℕ .
Which number is not a rational number?
A.
0.3
Terminating
Decimal
B.
𝟐
𝟓
Terminating
Decimal
C.
𝟑
-1
𝟒
Terminating
Decimal
D.
E.
F.
𝟏𝟔
0.𝟏
Terminating
Decimal
Recurring
Decimal
𝟐 =1.41421356237..
Decimal
expansion that
can go on
forever
without
recurring
Which number is not a rational number?
The value of n for which 𝒏 is rational
A.
2
B.
3
C.
5
D.
4
How many rational numbers are there between
0 and 1?
A.
100
B.
10
C.
Infinitely many
D.
5
Answer True or False to the following:
‘All rational numbers are a subset of the integers’.
TRUE
FALSE
Consider whether the following statements
are True or False?
Statement
True or False
Every integer is a natural number
False
Every natural number is a rational number
True
Every rational number is an integer
False
Every integer is a rational number
True
Every natural number is an integer
True
Which of the following venn-diagrams is correct?
ℤ
ℚ
ℕ
A.
B.
C.
ℚ
ℚ
ℤ
ℕ
ℕ
Natural
ℕ ℤ
ℕ and ℤ.
Page 23
Venn Diagram & Number Line
Natural
ℕ
Integers
ℤ
ℕ and ℤ.
Page 23
Venn Diagram & Number Line
ℤ
A.
ℚ\ℕ
B.
ℕ\ℤ
C.
ℤ\ℕ
ℕ
Page 23
Which symbol can we use for the ‘grey ‘ part of the
Venn-diagram?
Consider whether the following statement is
Always, Sometimes or Never True
‘An integer is a whole number.’
Always
Consider whether the following statement is
Always, Sometimes or Never True
‘Negative numbers are Natural numbers.’
Never
Consider whether the following statement is
Always, Sometimes or Never True
‘The square of a number is greater than that number’
Sometimes
Natural
(N)
Natural numbers
(ℕ)
&Numbers
Integers
(ℤ)
Number
Systems
Natural numbers (ℕ) : The natural numbers is the set of
counting numbers.
ℕ = 1, 2, 3, 4, 5, … . . .
The natural numbers is the set of positive whole numbers.
This set does not include the number 0.
Integers (ℤ) : The set of integers is the set of all whole
numbers, positive negative and zero.
ℤ = … . . −3, −2, −1, 0, 1, 2, 3, … . . .
Page 23
Summary
Rational Numbers (ℚ)
A Rational number(ℚ) is a number that can be written
𝑝
as a ratio of two integers , where p, q ∈ ℤ & q≠ 0.
𝑞
A Rational number will have a decimal expansion that
is terminating or recurring.
Examples:
a)
b)
c)
1
0.25 is rational , because it can be written as the ratio .
4
3
1.5 is rational , because it can be written as the ratio .
2
1
0.3 is rational , because it can be written as the ratio .
3
Interesting Rational Numbers
𝟓𝟑
𝟖𝟑
𝟏
𝟏
= 𝟎. 𝟏428𝟕
𝟕
=
𝟎.
𝟏𝟒𝟐𝟖𝟓𝟕𝟏𝟒𝟐𝟖𝟓𝟕𝟏𝟒𝟐𝟖𝟓𝟕
𝟎.
𝟕 𝟔385542168674698795180722
891566265060240𝟗
Literacy Considerations
•
•
•
•
•
•
•
•
Word Bank
Natural number
Integer
Rational number
Ratio
Whole Number
Recurring/Repeating decimal
Terminating decimal
Subset
ℕ, ℤ and ℚ.
Page 23
Venn Diagram & Number Line
Natural
ℕ
Integers
ℤ
ℤ\ℕ
ℕ, ℤ and ℚ.
Page 23
Venn Diagram & Number Line
ℕ, ℤ and ℚ.
Page 23
Venn Diagram & Number Line
Rational
ℚ
ℚ\ℤ
Rational
ℚ
𝟑
𝟐
𝟒
𝟐
𝟓
𝟐
𝟔
𝟐
Page 23
𝟐
𝟐
Rational
ℚ
Page 23
𝟒 𝟓 𝟔 𝟕 𝟖 𝟗 𝟏𝟎 𝟏𝟏 𝟏𝟐
𝟒𝟒 𝟒 𝟒 𝟒𝟒 𝟒 𝟒 𝟒
Rational
ℚ
𝟓
𝟒
𝟔
𝟒
𝟕
𝟒
𝟖
𝟒
Page 23
𝟒
𝟒
Rational
ℚ
Page 23
𝟖 𝟗 𝟏𝟎𝟏𝟏𝟏𝟐𝟏𝟑𝟏𝟒 𝟏𝟓𝟏𝟔
𝟖𝟖 𝟖 𝟖𝟖 𝟖𝟖 𝟖 𝟖
Rational
ℚ
𝟏𝟎
𝟖
𝟏𝟐
𝟖
Page 23
𝟖
𝟖
Rational
ℚ
𝟐𝟐 𝟐𝟒
𝟏𝟔 𝟏𝟔
Page 23
𝟏𝟔 𝟏𝟖 𝟐𝟎
𝟏𝟔 𝟏𝟔 𝟏𝟔
𝟐𝟐 𝟐𝟒
𝟏𝟔 𝟏𝟔
2
Rational
ℚ
Page 23
𝟏𝟔 𝟏𝟖 𝟐𝟎
𝟏𝟔 𝟏𝟔 𝟏𝟔
Learning
Outcomes
Number
Systems
Extend knowledge of number
systems from first year to include:
Within
Curriculum
Future
Strands
• IrrationalSubjects
numbers
• Surds
Across
Real
• Past
Real number system
Strands
World
Junior Certificate-All Levels
Leaving Certificate- Ordinary & Higher Level
Student Activity 1
Calculator Activity
Number
(1)
(2)
4
9
100
(3)
4
(4)
25
9
36
(5)
2
(6)
8
(7)
3
5
(8)
𝜋
(9)
1- 2
Calculator/ Decimals
Student Activity 1
Calculator Activity
Number
(1)
(2)
4
9
100
(3)
4
(4)
25
9
36
(5)
2
(6)
8
(7)
3
5
(8)
𝜋
(9)
1- 2
Calculator/ Decimals
2
Rational
3
0.3
Terminating
10
Or
2
0.6
Recurring
3
5
0.83
6
Irrational
1.414213562....2
1.41421356237309504….
Decimal
2.828427125….
2.82842712474619009….
2 2
expansion
1.70997594667669681….
1.709975947….
1.709975947 that can go
on forever
without
3.14159265358979323….
3.141592654….
𝜋
recurring
-0.41421356237497912.…
-0.4142135624…
1- 2
Irrational Numbers
An Irrational number is any number that cannot be
𝑝
expressed as a ratio of two integers , where p and q ∈ ℤ
𝑞
and q≠0.
Irrational numbers are numbers that can be written as
decimals that go on forever without recurring.
Page 23
So some numbers cannot be written as a ratio of two
integers…….
What is a Surd?
A Surd is an irrational
number containing a root term.
Number
4
9
100
4
25
3
9
36
Calculator/
Decimals
2
0.3
2
3
5
0.83
6
0.6
2
1.414213562
2
8
2.828427125
2√2
5
1.709975947
𝜋
1- 2
3.141592654
𝜋
-0.4142135624
1- 2
Irrational
Surd
Best known Irrational Numbers
Famous Irrational Numbers
𝝅
Pi : The first digits look like this
3.1415926535897932384626433832795……
𝒆
Euler’s Number: The first digits look like this
2.7182818284590452353602874713527….
𝝋
The Golden Ratio: The first digits look like
this: 1.6180339887498948420…….
√𝟐
Many square roots, cube roots, etc are also
irrational numbers.
𝟐 = 1.4142135623746……
𝟐
Pythagoras
Hippassus
1.4142135623746……
Irrational Numbers
Familiar
irrationals
3
5
7𝑒
Rational
ℚ
Are these the only irrational numbers
based on these numbers?
𝜋
Page 23
2
𝟐
𝟐
𝟕
𝟐
Rational
𝟑
𝟓
ℚ
𝟐
𝟐
𝒆
𝟐
𝝅
𝟐
3
5
7𝑒
𝜋
Page 23
2
Rational
ℚ
7𝑒 𝜋
2 2 2
Page 23
5
2
Rational
ℚ
7𝑒 𝜋
2 2 2
Page 23
5
2
Learning Outcomes
Extend knowledge of number
systems from first year to include:
• Irrational numbers
• Surds
• Real number system
Real Number System (ℝ)
The set of Rational and Irrational numbers
together make up the Real number system (ℝ).
Real Number System (ℝ)
Real
ℝ
𝟖
𝟑+ 𝟓
Rational
ℚ
Irrational Numbers
ℝ\ℚ
Type equation here.
− 𝟏𝟏
𝝅
𝟏− 𝟐
Student Activity
Classify all the following numbers as natural, integer, rational,
irrational or real using the table below. List all that apply.
5
Natural
ℕ
Integer
ℤ
Rational
ℚ



Irrational
ℝ\ℚ
Real
ℝ

1+ 2


−9.6403915 …


1
−2


6.36


2

-3
3
8
0
- 3













Now place these numbers as accurately as possible on
the number line below.
Now place them as accurately as possible on the number
What would help us here?
line below.
-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
The diagram represents the sets: Natural Numbers ℕ,
Integers ℤ, Rational Numbers ℚ 𝑎𝑛𝑑 Real Numbers ℝ.
Insert each of the following numbers in the correct place on the
diagram:
1
2
3
5, 1 + 2, −9.6403915. . … , − , 6.36 , 2𝜋, -3, 8, 0 and - 3.
ℝ
ℚ
ℚ
ℤ
ℕ
ℕ
The diagram represents the sets: Natural Numbers ℕ,
Integers ℤ, Rational Numbers ℚ 𝑎𝑛𝑑 Real Numbers ℝ.
Insert each of the following numbers in the correct place on the
diagram:
𝟏
−𝟗. 𝟔𝟒𝟎𝟑𝟗𝟏𝟓. . … , −
𝟐
5, 𝟏 + 𝟐,
ℝ
1+ 𝟐
ℚ
6.𝟑𝟔
𝟐𝝅
- 𝟑
, 6.𝟑𝟔, 2𝝅 , -3,
ℤ
𝟏
−
𝟐
ℕ
-3
0
-9.6403915…
𝟑
𝟑
𝟖
𝟖 , 0 and - 𝟑.
5
Session 2
Investigating Surds
Pythagoras
Hippassus
Show that
8 + 18 = 50 without the use of a calculator.
Show that
8
8 + 18 = 50 without the use of a calculator.
+
18
50
4x2 +
9x2
25 x 2
4 2 +
9 2
25 2
2 2
3 2
5 2
+
5 2
⇒ 8
+
18
=
50
Investigating Surds
Prior Knowledge
• Number Systems
(ℕ, ℤ ,ℚ, ℝ\ℚ & ℝ).
• Trigonometry
• Geometry/Theorems
• Co-ordinate Geometry
• Algebra
Investigating Surds
Plot A (0,0), B (1,1) &
C (1,0) and join them.
Write and Wipe
Desk Mats
Taking Formula
Length
a closer look
(Distance)
at surds graphically
𝑨𝑩 = Plot𝒙𝟐A−(0,0),
𝒙𝟏 𝟐 +
𝒚𝟐 −&𝒚𝟏
B (1,1)
𝟐 +join
𝑨𝑩 =C (1,0)
𝟏 − 𝟎and
𝟏 −them.
𝟎 𝟐
𝒚
𝑩 (𝟏, 𝟏)
(𝒙𝟐 ,𝒚𝟐 )
𝑨𝑩 =
𝟐
𝟏 𝟐Find
+ 𝟏|𝑨𝑩|
𝑨𝑩 = 𝟏 + 𝟏
𝑨
(𝟎, 𝟎)
(𝒙𝟏 ,𝒚𝟏 )
𝒙
𝑪
(1,0)
𝑨𝑩 = 𝟐
𝟐
Taking a closer
Pythagoras’
Theorem
look at surds graphically
𝒄𝟐 = 𝒂² + 𝒃²
𝒄𝟐 = 𝟏² + 𝟏²
𝒚
𝒄𝟐 = 𝟏 + 𝟏
𝒄
𝟏 𝒃
𝟏
𝒂
𝒄𝟐 = 𝟐
𝒙
𝒄𝟐 = 𝟐
𝒄 =
𝟐
Investigating Surds
1. Plot D (2,2) and E (2,0).
2. Join (1,1) to (2,2) and
join (2,2) to (2,0).
Write and Wipe
Desk Mats
Taking a closer
Pythagoras’
Theorem
look at surds graphically
𝑫
𝒄
𝟐
?
𝟖
2 𝒂
𝑩
2
𝒃
2.𝒄𝟐 Join
= 𝟐²(1,1)
+ 𝟐²to (2,2) and
join (2,2) to (2,0).
3.𝒄𝟐 Find
= 𝟒 |𝑨𝑫|
+𝟒
𝒄𝟐 = 𝟖
𝒄𝟐 = 𝟖
𝑬
𝑨
𝟐
𝒄
= 𝒂²
𝒃² and E (2,0).
1. Plot
D+
(2,2)
𝒄 =
𝟖
(1)
Length Formula (Distance)
(𝒙𝟐 ,𝒚𝟐 )
(2, 2)
(𝒙𝟏 ,𝒚𝟏 )
𝟐
𝑫
𝑨𝑩 =
(𝒙𝟐 −𝒙𝟏 )²+ (𝒚𝟐 −𝒚𝟏 )²
𝑨𝑩 =
(2− 1)² + ( 2− 1)²
(1, 𝟏)
𝑩
𝑨𝑩 =
(1)² + (1)²
𝑨𝑩 =
1 +
|AB| =
𝟐
1
(2) Pythagoras’ Theorem
D
𝑩
𝑪
𝒄𝟐 = 𝒂² + 𝒃²
𝑩𝑫
𝟐
= 𝟏² + 𝟏²
𝑩𝑫
𝟐
=𝟏+𝟏
𝑩𝑫
𝟐
=𝟐
𝑩𝑫
𝟐
= 𝟐
𝑩𝑫 =
𝟐
(2)
Pythagoras’ Theorem
𝒄
𝒄𝟐 = a² + b²
𝟐
1 𝒂
1
𝒃
𝒄𝟐 =1²+ 1²
𝒄𝟐 = 1 + 1
𝒄𝟐 = 2
𝒄𝟐 =
𝟐
c =
𝟐
(3) Congruent Triangles
SAS
𝟐
𝟐
1
1
1
1
Two sides
and the
included
angle
(4) Similar Triangles
𝑥
45°
𝒙
1
45°
𝟐
45°
1
45°
1
1
=
2 1
1
𝑥
2
=1
𝑥
1
2.
= . 2
2 1
𝑥= 2
𝑎
sin 𝜃 =
𝑐
𝒙
𝒄
sin 450
𝒂
1
45°
𝑥. sin 450
1𝒃
1
=
𝑥
1
= .𝑥
𝑥
𝑥 sin 450 = 1
𝑎
sin 𝜃 =
𝑐
𝑏
cos 𝜃 =
𝑐
𝑎
tan 𝜃 =
𝑏
𝑥 sin 450
1
=
0
sin 45
sin 450
𝑥= 2
Page 16
(5) Trigonometry
What are the possible misconceptions with
Multiplication of surds
𝟐+ 𝟐?
Graphically
𝟖
𝟐
𝟐
𝟖= 𝟐+ 𝟐
𝟖=𝟐 𝟐
Algebraically
𝟖=𝟐 𝟐
𝟖= 𝟒 𝟐
𝟒𝐱𝟐= 𝟒 𝟐
𝒂𝒃= 𝒂 𝒃
Division of Surds
Graphically
𝟖
𝟐
𝟐
𝟖
𝟐
= 2 or
𝟖
𝟐
=
𝟐 𝟐
𝟐
Algebraically
𝟖
𝟐
𝟒𝟖𝐱 𝟐
==
= 𝟒=𝟐
𝟐𝟐
𝟒 𝟐𝒂
𝒂
=
=𝟐
𝒃
𝒃
= 𝟒
=𝟐
=𝟐
Student Activity-White Board
Continue using the same whiteboard:
(1) Plot (3,3).
(2) Join (2,2) to (3,3) and join (3,3) to (3,0).
(3) Using (0,0), (3,0) and (3,3) as your triangle verify that the
length of the hypotenuse of this triangle is 18.
(4) Simplify 18 without the use of a calculator.
(5) Simplify
18
2
(6) Simplify
18
without
8
without the use of a calculator.
the use of a calculator.
Q1,2 &3
𝒄𝟐 = a² + b²
c
𝒄𝟐 = 3²+ 3²
𝟏𝟖
3
3
b
a
𝒄𝟐 = 9 + 9
𝒄𝟐 = 18
𝒄𝟐 =
𝟏𝟖
c =
𝟏𝟖
Q4
Simplify 18 without the use of a calculator.
Graphically
√2
𝟏𝟖 =
𝟏𝟖
√2
√2
3
𝟏𝟖 =
𝟐+ 𝟐+ 𝟐
𝟑 𝟐
Algebraically
𝒂𝒃
3
𝟏𝟖
=
𝒂 𝒃
=
𝟗
𝟐
𝟏𝟖 = 𝟑 𝟐
Q5.
Simplify
18
2
without the use of a calculator.
Graphically
𝟏𝟖
𝟐
𝟏𝟖
𝟐
𝟐
=3
Algebraically
𝟐
𝟏𝟖
𝟐
=
=
𝒂
𝒃
=
𝟗
𝟏𝟖𝐱 𝟐
𝟐𝟐
𝟗
=𝟐
𝟗
= 3
= 𝟗=3
𝟐
𝒂
𝒃
Q6.
Simplify
18
8
without the use of a calculator.
Graphically
𝟏𝟖
𝟖
𝟏𝟖
𝟖
=
𝟑
𝟐
𝟏𝟖
𝟖
or
=
𝟑 𝟐
𝟐 𝟐
=
𝟑
𝟐
Algebraically
𝟏𝟖
𝟖
=
=
𝟗𝟏𝟖𝐱 𝟐
𝟖
=
𝟒𝐱𝟐
𝟗
𝟗𝐱𝟐
=
𝟒𝐱𝟐
𝟐
𝒂𝟒 𝟐 𝒂
𝟗=
𝒃
=𝒃
𝟒
=
𝟑
𝟐
𝟗
𝟒
𝟑
=
𝟐
What other surds could we illustrate if we extended this diagram ?
𝟐𝟐 =1=1𝟐 𝟐
𝟖𝟖 =2=2𝟐 𝟐
𝟏𝟖𝟏𝟖
=3=3𝟐 𝟐
𝟑𝟐=4=4𝟐 𝟐
𝟐
𝟑𝟐
𝟐
𝟐
𝟐
1
1
2
2
34
3
4
What other surds could we illustrate if we extended this diagram ?
√2 =1√2
√8 =2√2
√18 =3√2
√32 =4√2
𝟓𝟎
𝟐
𝟐
𝟐
5
√50 = 5√2
𝟐
𝟐
5
√72 = 6√2
√98 = 7√2
√128 = 8√2
√162 = 9√2
√200 =10√2
Division of Surds
Graphically
𝟓𝟎
𝟑𝟐
𝟓𝟎
𝟑𝟐
=
𝟓
𝟒
Algebraically
𝟓𝟎
𝟑𝟐
=
𝟐𝟓 𝑿 𝟐
=
𝟓
=
𝟒
𝟏𝟔 𝑿 𝟐
𝟐𝟓
𝟐
𝟏𝟔
𝟐
Show that
8
8 + 18 = 50 without the use of a calculator.
+
18
50
4x2 +
9x2
25 x 2
4 2 +
9 2
25 2
2 2
3 2
5 2
+
5 2
⇒ 8
+
18
=
50
𝟖 + 𝟏𝟖
2 𝟖𝟐
𝟑𝟏𝟖
𝟐
3
2
3
2
𝟖 + 𝟏𝟖
= 2 𝟐 +𝟑 𝟐
=
5 𝟐
𝟖 + 2𝟏𝟖𝟐 =
= 𝟓 𝟓𝟎
+ 𝟓𝟑 𝟐
𝟐=
𝟐
𝟓𝟎
Pythagoras
Theorem
𝟐
𝟐
𝟓 𝟐𝟐
𝟓
𝟐
𝟐
𝟓
𝟖 + 𝟏𝟖 =
𝟓𝟎
𝟓
𝒄𝟐 = a² + b²
𝒄𝟐 = 2²+ 1²
b
𝟓
1
c
a
2
𝒄𝟐 = 4 + 1
𝒄𝟐 = 5
𝒄² = 𝟓
c = 𝟓
𝟑
𝒄𝟐 = a² + b²
𝟑
√3
11
𝒄𝟐 =( 𝟐)²+ 1²
𝒄𝟐 = 2 + 1
𝟐
𝒄𝟐 = 3
𝒄𝟐 =
𝟑
c = 𝟑
𝟓
𝟐𝟎
𝟓
2
𝟓
4
Graphically
𝟐𝟎
=
𝟓+ 𝟓
𝟐𝟎
= 𝟐 𝟓
Algebraically
𝟐𝟎
𝟐𝟎
= 𝟐 𝟓
= 𝟒
𝟓
𝟒𝑿𝟓 = 𝟒
𝟓
𝒂𝒃 = 𝒂
𝒃
𝟓
𝟒𝟓
𝟓
𝟓
𝟓
3
√45
6
Graphically
𝟒𝟓
𝟒𝟓
=
𝟓 + 𝟓+ 𝟓
= 𝟑 𝟓
Algebraically
𝟒𝟓
𝟒𝟓
= 𝟑 𝟓
= 𝟗
𝟓
𝟗𝑿𝟓 = 𝟗
𝟓
𝒂𝒃 = 𝒂
𝒃
Division of Surds
Graphically
𝟒𝟓
=3
𝟓
𝟒𝟓
𝟓
Algebraically.
𝟒𝟓
𝟓
=
=
=
𝒂
𝟒𝟓 𝑿 𝟓
𝟗
𝟓
= 𝟗=3
𝟓
𝟗
𝟓
𝟓
𝒂
== 𝟗
𝒃
𝒃
= 3
1
1
1
𝟓
𝟒
𝟑
𝟔
1
The Spiral of
Theodorus
1
1
𝟐
𝟕
𝟏𝟖
𝟖
1
𝟏𝟕
𝟗
1
1
1
𝟏𝟔
𝟏𝟎
𝟏𝟓
𝟏𝟏
𝟏𝟐
1
1
1
𝟏𝟑
1
𝟏𝟒
1
1
1
An Appreciation for students
 For positive real numbers a and b:
•
𝑎𝑏 = 𝑎 𝑏
•
𝑎
𝑏
=
𝑎
𝑏
 Adding /Subtracting Like Surds
 Simplifying Surds
Spiral Staircase Problem
Each step in a science museum's spiral staircase is an
isosceles right triangle whose leg matches the
hypotenuse of the previous step, as shown in the
overhead view of the staircase. If the first step has an
area of 0.5 square feet, what is the area of the eleventh
step?
Prior Knowledge
1
2
Area of a triangle= ah
Solution
Step 1
1
2
Step 2
1
b=
2
a
a² = 1
⇒𝑎=1
𝟐
1
2
𝟖
𝟐
2²=1
1
2
2 Multiplied by 2
(2)²=2
1
1
Area= sq.foot
2
Area= 1sq.foot
1
Step 1=
2
Step 2 = 1
Step 3 = 2
2
𝟐
1
2
Area
Step 3
Area= 2sq.feet
Step 4 =4
Step 5 =8
Step 6 =16
Step 7 =32
Step 8 =64
Step 9 =128
Step 10 =256
Step 11=512
Area
(11th Step)
512sq.feet
Solution
512 square feet. Using the area of a triangle formula, the first
step's legs are each 1 foot long. Use the Pythagorean theorem
to determine the hypotenuse of each step, which in turn is the
leg of the next step. Successive Pythagorean calculations
show that the legs double in length every second step: step 3
has 2-foot legs, step 5 has 4-foot legs, step 7 has 8-foot legs,
and so on. Thus, step 11 has 32-foot legs, making a triangle
with area 0.5(32)² = 512 sq. ft. Alternatively, students might
recognize that each step can be cut in half to make two copies
of the previous step. Hence, the area double with each new
step, giving an area of 512 square feet by the eleventh step.