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
Assessment
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).
The Integers are……
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.
Answer Trueor Falseto the following:
‘The natural numbers are a subset of the integers’.
TRUE
TRUE
FALSE
Which number is not an integer?
A. 1
B.0
C.
D. 4.
D.4.
The Rational numbers are…….
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.
Which number is not a rational number?
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.
Which number is not a rational number?
The value of n for which is rational
A.2
B.3
C. 5
D.4
D.4
D. 4
How many rational numbers are there between 0 and 1?
A. 100
B.10
C. Infinitely many
C.Infinitely many
D. 5
Answer Trueor False to the following:
‘All rational numbers are a subset of the integers’.
TRUE
FALSE
Consider whether the following statements are True or False?
False
True
False
True
True
Which of the following venndiagrams is correct?
ℕ
A.
B.
ℕ
Natural
ℕ
C.
ℕ
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Natural
ℕ
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ℤ ℕ
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A.ℚ\ℕ
B.ℕ\ℤ
C.ℤ\ℕ
C. ℤ\ℕ
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
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.
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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
385542168674698795180722891566265060240
428
Literacy Considerations Word Bank
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Natural
ℕ
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2
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Learning Outcomes
Within Strands
Curriculum
Subjects
Future
Across Strands
Real
World
Past
Student Activity 1Calculator Activity
Student Activity 1Calculator Activity
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…
Irrational Numbers
So some numbers cannot be written as a ratio of two integers…….
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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.
0.
0.8
1.414213562
2.828427125
3.141592654
0.4142135624
1
Best known Irrational Numbers
46……
Pythagoras
Hippassus
Familiar irrationals
Irrational Numbers
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Are these the only irrational numbers
based on these numbers?
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Learning Outcomes
The set of Rational and Irrational numbers together make up the Real number system (ℝ).
Real Number System (ℝ)
Student Activity
Classify all the following numbers as natural, integer, rational, irrational or real using the table below. List all that apply.
What would help us here?
10
2.5
0
2.5
5
7.5
7.5
10
5
The diagram represents the sets: Natural Numbers Integers RationalNumbersReal Numbers ℝ. Insert each of the following numbers in the correct place on the diagram:5, 6. , 2, 3, , 0 and 
ℕ
ℕ
The diagram represents the sets: Natural Numbers Integers RationalNumbersReal Numbers ℝ. Insert each of the following numbers in the correct place on the diagram:5, 6., 2 , 3, , 0 and 
1+
ℕ
6.
3
5
0

9.6403915…
Hippassus
Pythagoras
Show that = without the use of a calculator.
Show that = without the use of a calculator.
2 3
5
=
Investigating Surds
Plot A (0,0), B (1,1) &
C (1,0) and join them.
Write and Wipe
Desk Mats
Taking a closer look at surds graphically
Length Formula (Distance)
Plot A (0,0), B (1,1) &
C (1,0) and join them.
Find
Taking a closer look at surds graphically
Pythagoras’ Theorem
Investigating Surds
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
Taking a closer look at surds graphically
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
(1) Length Formula (Distance)
(2, 2
(1,
AB =
(2) Pythagoras’ Theorem
D
(2) Pythagoras’ Theorem
= a² + b²
=1²+ 1²
1
= 1 + 1
1
= 2
=
c =
(3) Congruent Triangles
SAS
Two sides and the included angle
1
1
1
1
(4) Similar Triangles
45°
1
45°
1
45°
1
45°
1
(5) Trigonometry
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1
45°
1
What are the possible misconceptions with
?
Multiplication of surds
Graphically
Algebraically
Division of Surds
Graphically
= 2
Algebraically
=
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
= a² + b²
c
= 3²+ 3²
= 9 + 9
3
a
= 18
=
3
c =
b
Graphically
√2
= + +
√2
=
3
√2
Algebraically
=
3
=
=
Graphically
= 3
Algebraically
= = 3
=
= 3
Graphically
=
=
=
or
Algebraically
= = =
=
=
=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
Graphically
=
Algebraically
Show that = without the use of a calculator.
2 3
5
=
2
3
2
3
2
+
=5
Pythagoras Theorem
=
=
= a² + b²
= 2²+ 1²
b
= 4 + 1
1
= 5
c
a
=
2
c =
= a² + b²
=)²+ 1²
1
1
√3
= 2 + 1
= 3
=
c =
2
4
Algebraically
Graphically
=
= +
=
=
=
=
3
√45
6
Graphically
Algebraically
=
= ++
=
=
=
=
Graphically
= 3
Algebraically.
= = 3
=
= 3
The Spiral of Theodorus
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
An Appreciation for students
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
Area of a triangle= ah
Solution
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
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 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.