Session1 Cultivating Skills for problem solving. Teaching the concept and notation of N umber S ystems using an understanding of basic rules and skills approach. Junior CertificateAll Levels. Leaving Certificate Foundation Level. Leaving Certificate Ordinary & Higher Level.
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Teaching the concept and notation of Number Systems
using an understanding of basic rules and skills approach.
Within Strands
Curriculum
Subjects
Future
Across Strands
Real
World
Past
Quiz Time…………
TheNatural numbers are…..
A. The set of all whole numbers , positive, negative and 0.
B. The set of all positive whole numbers (excluding 0).
B.The set of all positive whole numbers (excluding 0).
C. The set of all positive whole numbers (including 0).
A. The set of all whole numbers , positive, negative and 0.
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.
‘The natural numbers are a subset of the integers’.
TRUE
TRUE
FALSE
A. Any number of the form , where p, qℤ and q≠0.
A.Any number of the form , where p, qℤ and q≠0.
B.Any number of the form , where p, q ℤ.
C.Any number of the form , where p, q ℕ.
A.0.3
B.
Recurring
Decimal
Terminating
Decimal
Terminating
Decimal
Terminating
Decimal
Terminating
Decimal
Decimal expansion that can go on forever without recurring
C. 1
D.
E.0.
..
F.
F.
A. 100
B.10
C. Infinitely many
C.Infinitely many
D. 5
ℤ ℕ
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A. ℚ\ℕ
B. ℕ\ℤ
C. ℤ\ℕ
C. ℤ\ℕ
‘An integer is a whole number.’
Always
‘Negative numbers are Natural numbers.’
Never
‘The square of a number is greater than that number’
Sometimes
SummaryNumber Systems
Natural numbers (ℕ) & Integers (ℤ)
Natural numbers (ℕ) : The natural numbers is the set of counting numbers.
ℕ=
The natural numbers is the set of positive whole numbers. This set does not include the number 0.
Page 23
Integers (ℤ) : The set of integers is the set of all whole numbers, positive negative and zero.
ℤ=
A Rational number(ℚ) is a number that can be written as a ratio of two integers , where p, q ℤ & q≠ 0.
A Rational number will have a decimal expansion that is terminating or recurring.
Examples:
b) 1.5 is rational , because it can be written as the ratio
c) 0. is rational , because it can be written as the ratio
Within Strands
Curriculum
Subjects
Future
Across Strands
Real
World
Past
Rational
Terminating
Or
Recurring
0.
Decimal expansion that can go on forever without recurring
0.3
0.8
Irrational
….
1.414213562....
2.828427125….
2.82842712474619009….
1.709975947….
1.70997594667669681….
1.709975947
3.141592654….
3.14159265358979323….
3.141592654….
0.41421356237497912.…
0.4142135624…
So some numbers cannot be written as a ratio of two integers…….
Page 23
An Irrational number is any number that cannot be expressed as a ratio of two integers , where p and q
and q≠0.
Irrational numbers are numbers that can be written as
decimals that go on foreverwithout recurring.
A Surd is an irrational number containing a root term.
Irrational Numbers
Page 23
Are these the only irrational numbers
based on these numbers?
The set of Rational and Irrational numbers together make up the Real number system (ℝ).
Classify all the following numbers as natural, integer, rational, irrational or real using the table below. List all that apply.
10
2.5
0
2.5
5
7.5
7.5
10
5
ℕ
ℕ
1+
ℕ
6.
3
5
0

9.6403915…
Length Formula (Distance)
Plot A (0,0), B (1,1) &
C (1,0) and join them.
Find
Pythagoras’ Theorem
Plot D (2,2) and E (2,0).
Join (1,1) to (2,2) and join (2,2) to (2,0).
Write and Wipe
Desk Mats
Pythagoras’ Theorem
Plot D (2,2) and E (2,0).
Join (1,1) to (2,2) and join (2,2) to (2,0).
Find
?
2
2
Continue using the same whiteboard:
Plot (3,3).
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
(4) Simplify without the use of a calculator.
(5) Simplify without the use of a calculator.
(6) Simplify
=1
=2
=3
=4
=1
=2
=3
=1
=2
=3
=4
4
3
2
1
1
3
4
2
√2 =1√2
√8 =2√2
√18 =3√2
√32 =4√2
√50 = 5√2
= 5√2
√72 = 6√2
√98 = 7√2
√128 = 8√2
√162 = 9√2
√200 =10√2
5
5
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
Area of a triangle= ah
Area
Step 1=
Step 2 = 1
Step 3 = 2
Multiplied by 2
Step 4 =4
Step 5 =8
Step 6 =16
Step 7 =32
Step 8 =64
Step 9 =128
Step 10 =256
Step 11=512
Step 2
Step 1
Step 3
2
a b=
a² = 1
2
2
(2)²=2
²=1
1
1
Area= sq.foot
Area= 1sq.foot
Area= 2sq.feet
Area
(11th Step) 512sq.feet
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 2foot legs, step 5 has 4foot legs, step 7 has 8foot legs, and so on. Thus, step 11 has 32foot 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.