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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 Certificate-All Levels. Leaving Certificate- Foundation Level. Leaving Certificate- Ordinary & Higher Level.

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session1 cultivating skills for problem solving
Session1 Cultivating Skills for problem solving

Teaching the concept and notation of Number Systems

using an understanding of basic rules and skills approach.

section 1 number systems
Section 1 Number Systems

Within Strands

Curriculum

Subjects

Future

  • Prior Knowledge
  • Number Systems (ℕ, ℤ & ℚ)

Across Strands

Real

World

Past

slide6
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).

slide7
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.

slide8
Answer Trueor Falseto the following:

‘The natural numbers are a subset of the integers’.

TRUE

TRUE

FALSE

slide10
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.

slide11
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.

slide14
How many rational numbers are there between 0 and 1?

A. 100

B.10

C. Infinitely many

C.Infinitely many

D. 5

slide15
Answer Trueor False to the following:

‘All rational numbers are a subset of the integers’.

TRUE

FALSE

slide17
Which of the following venn-diagrams is correct?

A.

B.

Natural

C.

which symbol can we use for the grey part of the venn diagram
Which symbol can we use for the ‘grey ‘ part of the Venn-diagram?

ℤ ℕ

Page 23

A. ℚ\ℕ

B. ℕ\ℤ

C. ℤ\ℕ

C. ℤ\ℕ

slide21
Consider whether the following statement is Always, Sometimes or Never True

‘An integer is a whole number.’

Always

slide22
Consider whether the following statement is Always, Sometimes or Never True

‘Negative numbers are Natural numbers.’

Never

slide23
Consider whether the following statement is Always, Sometimes or Never True

‘The square of a number is greater than that number’

Sometimes

natural numbers n
Natural Numbers (N)

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=

rational numbers
Rational Numbers (ℚ)

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:

  • 0.25 is rational , because it can be written as the ratio

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

interesting rational numbers
Interesting Rational Numbers

385542168674698795180722891566265060240

428

slide27
Literacy Considerations Word Bank
  • Natural number
  • Integer
  • Rational number
  • Ratio
  • Whole Number
  • Recurring/Repeating decimal
  • Terminating decimal
  • Subset
slide31
Page 23
  • Rational
slide32
Page 23
  • Rational
slide33
Page 23
  • Rational
slide34
Page 23
  • Rational
slide35
Page 23
  • Rational
slide36
Page 23
  • Rational
slide37
2

Page 23

  • Rational
number systems
Learning OutcomesNumber Systems
  • Extend knowledge of number systems from first year to include:
  • Irrational numbers
  • Surds
  • Real number system

Within Strands

Curriculum

Subjects

Future

Across Strands

Real

World

Past

slide42
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…

slide43
Irrational Numbers

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.

what is a surd
What is a Surd?

A Surd is an irrational number containing a root term.

slide45
0.

0.8

1.414213562

2.828427125

3.141592654

-0.4142135624

1-

slide46
Best known Irrational Numbers

46……

Pythagoras

Hippassus

slide47
Familiar irrationals

Irrational Numbers

Page 23

  • Rational

Are these the only irrational numbers

based on these numbers?

slide48
Page 23
  • Rational
slide49
Page 23
  • Rational
slide50
Page 23
  • Rational
slide51
Learning Outcomes
  • Extend knowledge of number systems from first year to include:
  • Irrational numbers
  • Surds
  • Real number system
real number system
Real Number System (ℝ)

The set of Rational and Irrational numbers together make up the Real number system (ℝ).

slide53
Real Number System (ℝ)
  • Real
  • Rational
  • Irrational Numbers
  • ℝ\ℚ
slide54
Student Activity

Classify all the following numbers as natural, integer, rational, irrational or real using the table below. List all that apply.

slide57
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 -

slide58
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+

6.

-3

5

0

-

-9.6403915…

session 2 investigating surds
Session 2Investigating Surds

Hippassus

Pythagoras

investigating surds
Investigating Surds
  • Prior Knowledge
  • Number Systems
  • (ℕ, ℤ ,ℚ, ℝ\ℚ & ℝ).
  • Trigonometry
  • Geometry/Theorems
  • Co-ordinate Geometry
  • Algebra
slide63
Investigating Surds

Plot A (0,0), B (1,1) &

C (1,0) and join them.

Write and Wipe

Desk Mats

slide64
Taking a closer look at surds graphically

Length Formula (Distance)

Plot A (0,0), B (1,1) &

C (1,0) and join them.

Find

slide66
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

slide67
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

slide68
(1) Length Formula (Distance)

(2, 2

(1,

|AB| =

slide70
(2) Pythagoras’ Theorem

= a² + b²

=1²+ 1²

1

= 1 + 1

1

= 2

=

c =

slide71
(3) Congruent Triangles

SAS

Two sides and the included angle

1

1

1

1

slide72
(4) Similar Triangles

45°

1

45°

1

45°

1

45°

1

slide73
(5) Trigonometry

Page 16

1

45°

1

slide74
What are the possible misconceptions with

?

Multiplication of surds

Graphically

Algebraically

slide75
Division of Surds

Graphically

= 2

Algebraically

=

student activity white board
Student Activity-White Board

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

q1 2 3
Q1,2 &3

= a² + b²

c

= 3²+ 3²

= 9 + 9

3

a

= 18

=

3

c =

b

q4 simplify without the use of a calculator
Q4Simplify without the use of a calculator.

Graphically

√2

= + +

√2

=

3

√2

Algebraically

=

3

=

=

q5 simplify without the use of a calculator
Q5. Simplify without the use of a calculator.

Graphically

= 3

Algebraically

= = 3

=

= 3

q6 simplify without the use of a calculator
Q6. Simplify without the use of a calculator.

Graphically

=

=

=

or

Algebraically

= = =

=

=

what other surds could we illustrate if we extended this diagram
What other surds could we illustrate if we extended this diagram ?

=1

=2

=3

=4

=1

=2

=3

=1

=2

=3

=4

4

3

2

1

1

3

4

2

what other surds could we illustrate if we extended this diagram1
What other surds could we illustrate if we extended this diagram ?

√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

division of surds
Division of Surds

Graphically

=

Algebraically

slide85
2

3

2

3

2

+

=5

slide87
= a² + b²

= 2²+ 1²

b

= 4 + 1

1

= 5

c

a

=

2

c =

slide88
= a² + b²

=)²+ 1²

1

1

√3

= 2 + 1

= 3

=

c =

slide89
2

4

Algebraically

Graphically

=

= +

=

=

=

=

slide90
3

√45

6

Graphically

Algebraically

=

= ++

=

=

=

=

division of surds1
Division of Surds

Graphically

= 3

Algebraically.

= = 3

=

= 3

slide92
The Spiral of Theodorus

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

slide93
An Appreciation for students
  • For positive real numbers a and b:
  • =
  • Adding /Subtracting Like Surds
  • Simplifying Surds
slide94
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

slide95
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

slide96
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.

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