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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


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 venn-diagrams is correct?

A.

B.

Natural

C.


Venn diagram number line and
Venn Diagram & Number Line ℕ and ℤ.

Page 23

Natural


Venn diagram number line and1
Venn Diagram & Number Line ℕ and ℤ.

Page 23

  • Integers


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


Consider whether the following statement is Venn-diagram? Always, Sometimes or Never True

‘An integer is a whole number.’

Always


Consider whether the following statement is Venn-diagram? Always, Sometimes or Never True

‘Negative numbers are Natural numbers.’

Never


Consider whether the following statement is Venn-diagram? Always, Sometimes or Never True

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

Sometimes


Natural numbers n
Natural Numbers (N) Venn-diagram?

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 Venn-diagram? (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 Venn-diagram?

385542168674698795180722891566265060240

428


Literacy Considerations Word Bank Venn-diagram?

  • Natural number

  • Integer

  • Rational number

  • Ratio

  • Whole Number

  • Recurring/Repeating decimal

  • Terminating decimal

  • Subset


Venn diagram number line and2
Venn Diagram & Number Line Venn-diagram? ℕ,ℤ and ℚ.

Page 23

Natural


Venn diagram number line and3
Venn Diagram & Number Line Venn-diagram? ℕ,ℤ and ℚ.

Page 23

  • Integers

  • ℤ\ℕ


Venn diagram number line and4
Venn Diagram & Number Line Venn-diagram? ℕ,ℤ and ℚ.

Page 23

  • Rational

  • ℚ\ℤ


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


2 Venn-diagram?

Page 23

  • Rational


Number systems

Learning Outcomes Venn-diagram?

Number 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




Student Activity 1 Venn-diagram? Calculator Activity


Student Activity 1 Venn-diagram? Calculator 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 Venn-diagram?

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? Venn-diagram?

A Surd is an irrational number containing a root term.


0 Venn-diagram? .

0.8

1.414213562

2.828427125

3.141592654

-0.4142135624

1-


Best known Irrational Numbers Venn-diagram?

46……

Pythagoras

Hippassus


Familiar irrational Venn-diagram? s

Irrational Numbers

Page 23

  • Rational

Are these the only irrational numbers

based on these numbers?


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Page 23 Venn-diagram?

  • Rational


Learning Outcomes Venn-diagram?

  • Extend knowledge of number systems from first year to include:

  • Irrational numbers

  • Surds

  • Real number system


Real number system
Real Number System Venn-diagram? (ℝ)

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


Real Number System Venn-diagram? (ℝ)

  • Real

  • Rational

  • Irrational Numbers

  • ℝ\ℚ


Student Activity Venn-diagram?

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



Now place them as accurately as possible on the number line below

What would help us here? number line below.

Now place them as accurately as possible on the number line below.

10

2.5

0

-2.5

-5

7.5

-7.5

-10

5


The diagram represents the sets: Natural number line below.Numbers Integers RationalNumbersReal Numbers ℝ. Insert each of the following numbers in the correct place on the diagram:5, 6. , 2, -3, , 0 and -


The diagram represents the sets: Natural number line below.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 2 number line below.Investigating Surds

Hippassus

Pythagoras


Show that number line below. = without the use of a calculator.


Show that number line below. = without the use of a calculator.

2 3

5

=


Investigating surds
Investigating Surds number line below.

  • Prior Knowledge

  • Number Systems

  • (ℕ, ℤ ,ℚ, ℝ\ℚ & ℝ).

  • Trigonometry

  • Geometry/Theorems

  • Co-ordinate Geometry

  • Algebra


Investigating Surds number line below.

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 number line below.

Length Formula (Distance)

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

C (1,0) and join them.

Find


Taking a closer look at surds graphically number line below.

Pythagoras’ Theorem


Investigating Surds number line below.

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 number line below.

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) number line below.Length Formula (Distance)

(2, 2

(1,

|AB| =


(2) number line below.Pythagoras’ Theorem

D


(2) number line below.Pythagoras’ Theorem

= a² + b²

=1²+ 1²

1

= 1 + 1

1

= 2

=

c =


(3) Congruent Triangles number line below.

SAS

Two sides and the included angle

1

1

1

1


(4) Similar Triangles number line below.

45°

1

45°

1

45°

1

45°

1


(5 number line below.) Trigonometry

Page 16

1

45°

1


What are number line below.the possible misconceptions with

?

Multiplication of surds

Graphically

Algebraically


Division of Surds number line below.

Graphically

= 2

Algebraically

=


Student activity white board
Student Activity-White Board number line below.

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 number line below.

= a² + b²

c

= 3²+ 3²

= 9 + 9

3

a

= 18

=

3

c =

b


Q4 simplify without the use of a calculator
Q4 number line below.Simplify without the use of a calculator.

Graphically

√2

= + +

√2

=

3

√2

Algebraically

=

3

=

=


Q5 simplify without the use of a calculator
Q5. number line below.Simplify without the use of a calculator.

Graphically

= 3

Algebraically

= = 3

=

= 3


Q6 simplify without the use of a calculator
Q6. number line below.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 diagram ?

Graphically

=

Algebraically


Show that diagram ? = without the use of a calculator.

2 3

5

=


2 diagram ?

3

2

3

2

+

=5


Pythagoras Theorem diagram ?

=

+ =

=


= a² + b² diagram ?

= 2²+ 1²

b

= 4 + 1

1

= 5

c

a

=

2

c =


= a² + b² diagram ?

=)²+ 1²

1

1

√3

= 2 + 1

= 3

=

c =


2 diagram ?

4

Algebraically

Graphically

=

= +

=

=

=

=


3 diagram ?

√45

6

Graphically

Algebraically

=

= ++

=

=

=

=


Division of surds1
Division of Surds diagram ?

Graphically

= 3

Algebraically.

= = 3

=

= 3


The Spiral of diagram ?Theodorus

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1


An Appreciation for students diagram ?

  • For positive real numbers a and b:

  • =

  • Adding /Subtracting Like Surds

  • Simplifying Surds


Spiral Staircase Problem diagram ?

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 diagram ?

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 diagram ?

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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