Session1 cultivating skills for problem solving
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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.


Junior certificate all levels

Junior Certificate-All Levels


Leaving certificate foundation level

Leaving Certificate- Foundation Level


Leaving certificate ordinary higher level

Leaving Certificate- Ordinary & Higher Level


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


Session1 cultivating skills for problem solving

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).


Session1 cultivating skills for problem solving

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.


Session1 cultivating skills for problem solving

Answer Trueor Falseto the following:

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

TRUE

TRUE

FALSE


Session1 cultivating skills for problem solving

Which number is not an integer?

A. -1

B.0

C.

D. 4.

D.4.


Session1 cultivating skills for problem solving

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.


Session1 cultivating skills for problem solving

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.


Session1 cultivating skills for problem solving

Which number is not a rational number?


Session1 cultivating skills for problem solving

The value of n for which is rational

A.2

B.3

C. 5

D.4

D.4

D. 4


Session1 cultivating skills for problem solving

How many rational numbers are there between 0 and 1?

A. 100

B.10

C. Infinitely many

C.Infinitely many

D. 5


Session1 cultivating skills for problem solving

Answer Trueor False to the following:

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

TRUE

FALSE


Session1 cultivating skills for problem solving

Consider whether the following statements are True or False?

False

True

False

True

True


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

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

‘An integer is a whole number.’

Always


Session1 cultivating skills for problem solving

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

‘Negative numbers are Natural numbers.’

Never


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

Literacy Considerations Word Bank

  • Natural number

  • Integer

  • Rational number

  • Ratio

  • Whole Number

  • Recurring/Repeating decimal

  • Terminating decimal

  • Subset


Venn diagram number line and2

Venn Diagram & Number Line ℕ,ℤ and ℚ.

Page 23

Natural


Venn diagram number line and3

Venn Diagram & Number Line ℕ,ℤ and ℚ.

Page 23

  • Integers

  • ℤ\ℕ


Venn diagram number line and4

Venn Diagram & Number Line ℕ,ℤ and ℚ.

Page 23

  • Rational

  • ℚ\ℤ


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

2

Page 23

  • Rational


Number systems

Learning Outcomes

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


Junior certificate all levels1

Junior Certificate-All Levels


Leaving certificate ordinary higher level1

Leaving Certificate- Ordinary & Higher Level


Session1 cultivating skills for problem solving

Student Activity 1Calculator Activity


Session1 cultivating skills for problem solving

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…


Session1 cultivating skills for problem solving

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.


Session1 cultivating skills for problem solving

0.

0.8

1.414213562

2.828427125

3.141592654

-0.4142135624

1-


Session1 cultivating skills for problem solving

Best known Irrational Numbers

46……

Pythagoras

Hippassus


Session1 cultivating skills for problem solving

Familiar irrationals

Irrational Numbers

Page 23

  • Rational

Are these the only irrational numbers

based on these numbers?


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

Page 23

  • Rational


Session1 cultivating skills for problem solving

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).


Session1 cultivating skills for problem solving

Real Number System (ℝ)

  • Real

  • Rational

  • Irrational Numbers

  • ℝ\ℚ


Session1 cultivating skills for problem solving

Student Activity

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


Now place these numbers as accurately as possible on the number line below

Now place these numbers as accurately as possible on the number line below.


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

What would help us here?

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


Session1 cultivating skills for problem solving

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 -


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

Show that = without the use of a calculator.


Session1 cultivating skills for problem solving

Show that = without the use of a calculator.

2 3

5

=


Investigating surds

Investigating Surds

  • Prior Knowledge

  • Number Systems

  • (ℕ, ℤ ,ℚ, ℝ\ℚ & ℝ).

  • Trigonometry

  • Geometry/Theorems

  • Co-ordinate Geometry

  • Algebra


Session1 cultivating skills for problem solving

Investigating Surds

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

C (1,0) and join them.

Write and Wipe

Desk Mats


Session1 cultivating skills for problem solving

Taking a closer look at surds graphically

Length Formula (Distance)

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

C (1,0) and join them.

Find


Session1 cultivating skills for problem solving

Taking a closer look at surds graphically

Pythagoras’ Theorem


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

(1) Length Formula (Distance)

(2, 2

(1,

|AB| =


Session1 cultivating skills for problem solving

(2) Pythagoras’ Theorem

D


Session1 cultivating skills for problem solving

(2) Pythagoras’ Theorem

= a² + b²

=1²+ 1²

1

= 1 + 1

1

= 2

=

c =


Session1 cultivating skills for problem solving

(3) Congruent Triangles

SAS

Two sides and the included angle

1

1

1

1


Session1 cultivating skills for problem solving

(4) Similar Triangles

45°

1

45°

1

45°

1

45°

1


Session1 cultivating skills for problem solving

(5) Trigonometry

Page 16

1

45°

1


Session1 cultivating skills for problem solving

What are the possible misconceptions with

?

Multiplication of surds

Graphically

Algebraically


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

Show that = without the use of a calculator.

2 3

5

=


Session1 cultivating skills for problem solving

2

3

2

3

2

+

=5


Session1 cultivating skills for problem solving

Pythagoras Theorem

=

+ =

=


Session1 cultivating skills for problem solving

= a² + b²

= 2²+ 1²

b

= 4 + 1

1

= 5

c

a

=

2

c =


Session1 cultivating skills for problem solving

= a² + b²

=)²+ 1²

1

1

√3

= 2 + 1

= 3

=

c =


Session1 cultivating skills for problem solving

2

4

Algebraically

Graphically

=

= +

=

=

=

=


Session1 cultivating skills for problem solving

3

√45

6

Graphically

Algebraically

=

= ++

=

=

=

=


Division of surds1

Division of Surds

Graphically

= 3

Algebraically.

= = 3

=

= 3


Session1 cultivating skills for problem solving

The Spiral of Theodorus

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1


Session1 cultivating skills for problem solving

An Appreciation for students

  • For positive real numbers a and b:

  • =

  • Adding /Subtracting Like Surds

  • Simplifying Surds


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

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


Session1 cultivating skills for problem solving

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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