Cultivating Skills for problem solving Session1 Teaching the concept and notation of Number Systems using an understanding of basic rules and skills approach. Junior Certificate-All Levels Leaving Certificate- Foundation Level Leaving Certificate- Ordinary & Higher Level Section 1 Systems

Number Knowledge Within Prior Curriculum Future Strands Number Systems (, & Subjects ) Across Real Past Strands World Quiz Assessment

Time The Natural numbers are.. A. The set of all whole numbers , positive, negative and 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. B. The set of all positive whole numbers only. C. The set of all negative whole numbers only. Answer True or False to the following: The natural numbers are a subset of the integers. TRUE FALSE Which number is not an integer? A.

B. -1 0 C. D. 4. The Rational numbers are. A. Any number of the form , where p, q and q0. B. Any number of the form , where p, q .

C. Any number of the form , where p, q . Which number is not a rational number? Terminating Decimal A. 0.3 B. C. Terminating Decimal -1 Terminating

Decimal D. E. F. 2 Terminating Decimal 0. Recurring Decimal .. Decimal

expansion that can go on forever without recurring Which number is not a rational number? 2 The value of n for which is rational A. 2 B. 3 C. 5 D. 4 How many rational numbers are there between 0 and 1?

A. 100 B. 10 C. Infinitely many D. 5 Answer True or False to the following: All rational numbers are a subset of the integers. TRUE FALSE Consider whether the following statements are True or False? Statement True or False Every integer is a natural number

False Every natural number is a rational number True Every rational number is an integer False Every integer is a rational number Every natural number is an integer True True Which of the following venn-diagrams is correct?

A. B. C.

Natural Venn Diagram & Number Line Page 23 and . Natural Venn Diagram & Number Line Page 23

and . Integers A. \ B. \ C. \ \ Page 23 Which symbol can we use for the grey part of the Venn-diagram? 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

Natural numbers () : The natural numbers is the set of counting numbers. = The natural numbers is the set of positive whole numbers. This set does not include the number 0. Integers () : The set of integers is the set of all whole numbers, positive negative and zero. = Page 23 Summary Number Natural Natural numbers () &Numbers Integers (N)

() Systems Rational Numbers () A Rational number() is a number that can be written as a ratio of two integers , where p, q & q 0. A Rational number will have a decimal expansion that is terminating or recurring. Examples: a) 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 =.

428 385542168674698795180722891 566265060240 Literacy Considerations Word Bank Natural number Integer Rational number

Ratio Whole Number Recurring/Repeating decimal Terminating decimal Subset Venn Diagram & Number Line Page 23 and . , Natural Venn Diagram & Number Line

Page 23 and . , Integers \ Venn Diagram & Number Line Page 23 and . , Rational

\ Rational Page 23

Rational Page 23 Rational

Page 23 Rational Page 23

Rational Page 23 Rational

Page 23 2 Rational Page 23 Learning NumberOutcomes Systems Extend knowledge of number systems from first year to

Within Future include: Curriculum Strands Subjects Irrational numbers Across Real Surds Past Strands Real number system World Junior Certificate-All Levels

Leaving Certificate- Ordinary & Higher Level Student Activity 1 Calculator Activity Number (1) (2) (3) (4) (5) (6) (7) (8) (9) 1-

Calculator/ Decimals Student Activity 1 Calculator Activity Number (1) Calculator/ Decimals 2 (2) (2) (3) 0.3 (4)

(3) 0. (5) (6) (4) (7) (8) (5) (9) (6) (7) (8) (9) Rational Terminating Or

Recurring 2 0.8 1- 1.414213562.... . 1- Irrational Decimal 2.828427125. 2.82842712474619009. expansio n that 1.70997594667669681. 1.709975947.

1.709975947 can go on 3.14159265358979323.forever 3.141592654. without -0.41421356237497912. -0.4142135624 recurring 1.709975947 Numbers Irrational An Irrational number is any number that cannot be expressed as a ratio of two integers , where p and q and q0. Irrational numbers are numbers that can be written as

decimals that go on forever without recurring. Page 23 So some numbers cannot be written as a ratio of two integers. What is a Surd? A Surd is an irrational number containing a root term. Number Calculator/ Decimals 2 0.3 0.3

0. 2 0.8 1.709975947 1- 1.414213562 2.828427125 1.709975947 3.141592654 -0.4142135624 1- Irrational Surd

Best known Irrational Numbers Famous Famous Irrational Irrational Numbers Numbers Pi : The first digits look like this 3.1415926535897932384626433832795 Pi : The first digits look like this 3.1415926535897932384626433832795 Eulers Number: The first digits look like this 2.7182818284590452353602874713527. Eulers Number: The first digits look like this 2.7182818284590452353602874713527. The Golden Ratio: The first digits look like this: 1.6180339887498948420.

The Golden Ratio: The first digits look like this: 1.6180339887498948420. Many square roots, cube roots,Hippas etc are also Pythago irrational numbers. ras sus 46 46 Irrational Numbers Familiar irrationals Rational

Are these the only irrational numbers based on these numbers? Page 23 2 3 5 7 Rational

Page 23 2 3 5 7 5 7 Rational Page 23 2 2 2 2

5 7 Rational Page 23 2 22 2 Learning Outcomes Extend knowledge of number systems from first year to include:

Irrational numbers Surds Real number system Real Number System () The set of Rational and Irrational numbers together make up the Real number system (). Real Number System () Real +

Rational Irrational Numbers \ Student Activity Classify all the following numbers as natural, integer, rational, irrational or real using the table below. List all that apply. 5

Natural Integer Rational 6. 2 2

-3 -3 0 0 - Irrational \

Real

Now place these numbers as accurately as possible on the number line below. Now place them as accurately as possible on the What would help us here? number line below.

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10

The diagram represents the sets: Natural Numbers Integers Rational NumbersReal Numbers . . 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 Numbers

Integers Rational NumbersReal Numbers . . Insert each of the following numbers in the correct place on the diagram: 5 , 6., 2 , -3, , 0 and - 1+ 6. -

-3 0 -9.6403915 5 Session 2 Investigating Surds Pythagoras

Hippassus Show that = without the use of a calculator. Show that = without the use of a calculator. 50 4 x 2+ 9 x 2 4 25 x 2 2 25 2 2 + 9 2 3 5

= Investigating Surds Prior Knowledge Number Systems (, ,, \ & ). Trigonometry Geometry/Theorems Co-ordinate Geometry Algebra Investigating Surds Plot A (0,0), B (1,1) & C (1,0) and join them. Write and Wipe

Desk Mats Length Taking a Formula closer look (Distance) at surds graphically | |= ) +B ( (1,1) ( A (0,0), ) Plot ( , )

( , ) & | |= ( ) + ( ) C (1,0) and join

| |= ( ) them. +( ) | |= + Find ||= (1,0) Pythagoras Taking a closer Theorem look at surds graphically

= + = + =+ =

= = Investigating Surds 1. 1. Plot Plot D D (2,2) (2,2) and and E E (2,0). (2,0). 2. 2. Join Join (1,1) (1,1) to to (2,2) (2,2) and

and join join (2,2) (2,2) to to (2,0). (2,0). Write and Wipe Desk Mats Pythagoras Taking a closer Theorem look at surds graphically

? 2 2 Plot

= and 1. D 1. Plot+ D (2,2) (2,2) and E E (2,0). (2,0). = + 2. 2. Join Join (1,1) (1,1) to

to (2,2) (2,2) and and (2,2) join (2,2) to (2,0). (2,0). join =+ to 3. 3. Find Find = =

= (1) (Distance) Length Formula (1, (2, 2

| |= ||= (2 1) + ( 2 1) ||= (1) + (1) ||= |AB| = 1 + 1 (2) Pythagoras Theorem D

= + | | = + | | =+ | | = | | = | |= (2) Theorem Pythagoras = a + b

1 1 =1+ 1 =1+1 = 2 = c = (3) Congruent Triangles

SAS 1 1 1 1 Two sides and the included angle (4) Similar Triangles

45 1 45 45 1 45 1 1 Page 16

(5) Trigonometry 1 45 1 sin = cos = tan =

What are the possible misconceptions with Multiplication of surds ? Graphically Algebraically = = =

Division of Surds Graphically = 2 or = Algebraically = =

= Student Activity-White Board Continue using the same whiteboard:

(1) Plot (3,3). (2) 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 = a + b c = 3+ 3 3 3

b a =9+9 = 18 = c = Q4 Simplify without the use of a calculator. Graphically 2 2 2

= + + 3 = Algebraically = 3 = = Q5. Simplify without the use of a calculator. Graphically

=3 Algebraically == 3 = = 3 Q6. Simplify without the use of a

calculator. Graphically = or = Algebraically == =

= = = What What other other surds could we illustrate if we extended this diagram diagram ? =1 =2

=3 1 1 2 2 34 3 4 =1 =2 =3 =4

=4 What What other other surds could we illustrate if we extended this diagram diagram ? 2 =12 8 =22 18 =32 32 =42 50 = 52 5 5 72 = 62 98 = 72

128 = 82 162 = 92 200 =102 Division of Surds Graphically = Algebraically

Show that = without the use of a calculator. 50 4 x 2+ 9 x 2 4 25 x 2 2 25 2 2 + 9 2 3

5 = 2 3 2 3 2 + = 5

+ == = = a + b = 2+ 1 b 1 c a 2

=4+1 = 5 = c = = a + b 3 11 =)+ 1 = 2 +1

= 3 = c = 4 Graphically = + = 2

Algebraically = = = = 3 45

6 Graphically = ++ = Algebraically = = = =

Division of Surds Graphically =3 Algebraically. == 3

= = 3 1 1 1 1 1

1 The Spiral of Theodorus 1 1

1 1 1 1 1 1 1 1 1

An Appreciation for students For positive real numbers a and b: = Adding /Subtracting Like Surds Simplifying Surds 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 Step 1 a b= a = 1 = 1 Step 2 2 1 = 1=1

Area Step 3 Step 1= Step 2 = 1 Step 3 = 2 2 2 Multiplied by 2 (2) = 1=2 1 Area= sq.foot Area= 1sq.foot Area= 2sq.feet

Step 4 =4 Step 5 =8 Step 6 =16 Step 7 =32 Step 8 =64 Step 9 =128 Step 10 =256 Step 11=512 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 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) = 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.