PROBABILITY WORKSHOP with blocks By Jon - PowerPoint Presentation

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PROBABILITY WORKSHOP with blocks By Jon - PPT Presentation

PROBABILITY WORKSHOP with blocks By Jon Molomby Hands on L wr 2 ndry MATHS Q What is the waitress asking Review of Terms AND and OR ID: 762058

blocks permutations order red permutations blocks red order important green blue ans combinations yellow easier amp chosen hands pascal

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PROBABILITY WORKSHOP with blocks By Jon Molomby Hands -on Lwr. 2 nd’ry MATHS

Q : What is the waitress asking ?Review of Terms : AND and OR

The waitress : “ Tea OR Coffee ? ” Review of Terms : AND and OR

A mathematician might say : “ ‘Tea OR Coffee ?’ …

… means Tea OR Coffee OR BOTH ”

… but Tea AND Coffee looks like this.

- not BOTH ? ” Do you mean Tea XOR Coffee

The waitress : “ Yes , that’s what I mean ! Tea XOR Coffee ? ”

1. Permutations 2. Combinations3. Random Choice Green Blue Red Yellow We will look at PROBABILITY using 4 blocks download the worksheet from www.mathswithgeoboards.com You should have 4 blocks of different colour

1. PERMUTATIONS with Hands-on Lwr. 2 nd’ry MATHS

When Caleb Gattegno taught Permutations (in “ Mathematics at your Fingertips” : 1961 ) he would ask students to make different carriages for a train using Cuisenaire rods

Permutations Green No. of “carriages” No. of different arrangements ? Simply expressed1 (G )1 ( G )1 !

Permutations Green Blue No. of “carriages” No. of different arrangements ?Simply expressed1 (G )1 ( G )1 !2 (G & B )2 ( GB, BG )2 !

Permutations Green Blue Red No. of “carriages” No. of different arrangements ?Simply expressed1 (G )1 ( G ) 1 !2 (G & B )2 ( GB, BG )2 !3 (G & B & R )6 ( GBR , GRB , BRG , B G R , R B G , R G B ) 3 !

Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ?Simply expressed 1 (G )1 ( G )1 !2 (G & B )2 ( GB, BG )2 !3 (G & B & R ) 6 ( G BR, G R B , B R G , B G R , R B G , R G B ) 3 ! 4 ( G & B & R & Y ) 24 4 !

Permutations The General Rule n number of distinct things can be arranged inn ! ways

 PERMUTATIONS with 2) Four blocks (order important) ? Hint :(remember Q.1)

 PERMUTATIONS with 2) Four blocks (order important) ? 4

 PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3

 PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2

 PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1

 PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1 = 24

 PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1 = 24 Ans : 24

 PERMUTATIONS with 3) Red always first (order important) ?

 PERMUTATIONS with 3) Red always first (order important) ? 1 ? ? ? Red

 PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 ? ? Red

 PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 x 2 ? Red

 PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 x 2 x 1 = 6 Red Ans : 6

 PERMUTATIONS with 4) Red not first ? Hint :(remember Q.3)

 PERMUTATIONS with 4) Red not first ? All possibilities minus red first

 PERMUTATIONS with 4) Red not first ? All possibilities minus red first (from Q.3) 24 -

 PERMUTATIONS with 4) Red not first ? All possibilities minus red first (from Q.3) 24 - 6 =

 PERMUTATIONS with 4) Red not first ? All possibilities minus red first (from Q.3) 24 - 6 = 18 Ans : 18

 PERMUTATIONS with 5) Yellow first or last ?

 PERMUTATIONS with 5) Yellow first or last ? Mutually exclusive , so Yellow first plus Yellow last

 PERMUTATIONS with 5) Yellow first or last ? Mutually exclusive , so Yellow first plus Yellow last 6 + 6 = 12 Ans : 12

 PERMUTATIONS with 6) Yellow neither first nor last ? Hint :(remember Q.5)

 PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5)

 PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 -

 PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 - 12

 PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 - 12 = 12 Ans : 12

 PERMUTATIONS with 7) Red and Blue together ?

 PERMUTATIONS with 7) Red and Blue together ? R B

 PERMUTATIONS with 7) Red and Blue together ? 3 x 2 x 1 = 6 BUT RB BR R B so 6 x 2! Ans : 12

 PERMUTATIONS with 8) Red and Blue never together ? Hint :(remember Q.7)

 PERMUTATIONS with 8) Red and Blue never together ? All possibilities minus red and blue together (from Q.7)

 PERMUTATIONS with 8) Red and Blue never together ? All possibilities minus red and blue together (from Q.7) 24 - 12 = 12 Ans : 12

 PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Hint :(remember Q.7)

 PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB)

 PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB) i.e. GBRY , GYRB BR YG , Y RBG

 PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB) i.e. GBRY , GYRB BR YG , Y RBG 12 - 4 = 8 Ans : 8

 PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Checking by listing all permutations :RBGY RBYG RGBY RGYB RYBG RYGBBRGY BRYG BGRY BGYR BYRG BYGRGRBY GRYB GBRY GBYR GYRB GYBRYRBG YRGB YBRG YBGR YGRB YGBR Ans : 8

 PERMUTATIONS with 10) Red on one end AND Green the other

 PERMUTATIONS with 10) Red on one end AND Green the otherLike this : R _ _ G R _ _ G G _ _ R G _ _ R

 PERMUTATIONS with 10) Red on one end AND Green the otherPossibilities : R B Y G R Y B G G B Y R G Y B R Ans :

 PERMUTATIONS with 10) Red on one end AND Green the otherR B Y G R Y B G G B Y R G Y B R Ans : 4

 PERMUTATIONS with 10) Red on one end AND Green the otherChecking by listing all permutations :RBGY RBYG RGBY RGYB RYBG RYGBBRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBRYRBG YRGB YBRG YBGR YGRB YGBR Ans : 4

 PERMUTATIONS with 11) Red on one end OR Green on an end

 PERMUTATIONS with 11) Red on one end OR Green on an endMeans Red on one end OR Green on an end or BOTH

 PERMUTATIONS with 11) Red on one end OR Green on an endMeans Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle

 PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle i.e. _ RG _ _ GR _ _ RG _ _ GR _

 PERMUTATIONS with 11) Red on one end OR Green on an endMeans Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle possibilities B RG Y B GR Y Y RG B Y GR B

 PERMUTATIONS with 11) Red on one end OR Green on an endMeans Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle possibilities B RG Y B GR Y Y RG B Y GR B = 24 - 4 = 20 Ans : 20

 PERMUTATIONS with 11) Red on one end OR Green on an end or BOTHChecking by listing all permutations :RBGY RBYG RGBY RGYB RYBG RYGBBRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBRYRBG YRGB YBRG YBGR YGRB YGBR Ans : 20

 PERMUTATIONS with 12) Red on an end XOR Green on an end

 PERMUTATIONS with 12) Red on an end XOR Green on an endMeans Red on an end OR Green on an end minus Red on an end AND Green on an end

 PERMUTATIONS with 12) Red on an end XOR Green on an endMeans Red on an end OR Green on an end (Q.11) minus Red on an end AND Green on an end (Q.10)

 PERMUTATIONS with 12) Red on an end XOR Green on an endMeans Red on an end OR Green on an end (Q.11) minus Red on an end AND Green on an end (Q.10) 20 - 4 = 16 Ans : 16

 PERMUTATIONS with 12) Red on an end XOR Green on an endChecking by listing all permutations :RBGY RBYG RGBY RGYB RYBG RYGBBRGY BRYG BGRY BGYR BYRG BYGRGRBY GRYB GBRY GBYR GYRB GYBRYRBG YRGB YBRG YBGR YGRB YGBR Ans : 16

 PERMUTATIONS with 13) 4 Blocks in a circle

 PERMUTATIONS with 13) 4 Blocks in a circleFormula for things in a circle = ( n – 1 ) !

 PERMUTATIONS with 13) 4 Blocks in a circleFormula for things in a circle = ( n – 1 ) ! So the number of blocks minus 1 factorial = (4 – 1 )! = 3 ! Ans : 6

 PERMUTATIONS with 13) 4 Blocks in a circlethe number of blocks factorial minus 1 = (4 – 1 )! Checking by listing all permutations : Ans : 6

 PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other

 PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other All ways in a circle minus Red/Blue together

 PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other All ways in a circle minus Red/Blue together 3 ! - 2 ! 2 ! = 6 – 4 = 2 Ans : 2

 PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each otherChecking by listing all permutations : Let 1 = Red and 2 = Blue Ans : 2

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS :

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS : 4P2 =

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS : 4P2 = Ans : 12

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?An easier way ?

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?An easier way ? YesBased on nPk = n factorial to the first k no. of factors

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors 4 x 3 = 12

 PERMUTATIONS with 15) 2 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors 4 x 3 = 12 Ans : 12

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS :

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS : 4P3 =

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?Using the formula for PERMUTATIONS : 4P3 = Ans : 24

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?An easier way ?

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?An easier way ? YesBased on nPk = n factorial to the first k no. of factors

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors 4 x 3 x 2 =

 PERMUTATIONS with 16) 3 selected from 4 blocks (order important)?An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors 4 x 3 x 2 = 24 Ans : 24

2. COMBINATIONS with Hands-on Lwr. 2 nd’ry MATHS

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?Using the formula for Combinations :

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?Using the formula for Combinations : Ans : 6

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?An easier way ?

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?An easier way ? Yes … Using Pascal’s Triangle

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?An easier way ? Using Pascal’s Triangle

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?An easier way ? 4C2 Using Pascal’s Triangle

 COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)?An easier way ? 4C2 Using Pascal’s Triangle Ans : 6

 COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)?An easier way ? Using Pascal’s Triangle

 COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)?An easier way ? 4C3 Using Pascal’s Triangle

 COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)?An easier way ? 4C3 Using Pascal’s Triangle Ans : 4

 COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)?An easier way ? Using Pascal’s Triangle

 COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)?An easier way ? 4C4 Using Pascal’s Triangle

 COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)?An easier way ? 4C4 Using Pascal’s Triangle Ans : 1

 COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)?An easier way ? Using Pascal’s Triangle

 COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)?An easier way ? 4C0 Using Pascal’s Triangle

 COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)?An easier way ? 4C0 Using Pascal’s Triangle Ans : 1

 COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)?An easier way ? Using Pascal’s Triangle

 COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)?An easier way ? 4C1 Using Pascal’s Triangle

 COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)?An easier way ? 4C1 Using Pascal’s Triangle Ans : 4

 COMBINATIONS with Another handy trick with Combinations nCr = nCn-r e.g. 20C18 = 20C2

3. Random Choice without replacement Hands -on Lwr. 2 nd’ry MATHS

 RANDOM CHOICE with Put the 4 blocks inside your hands : What are the chances if taken at random …. ?

 RANDOM CHOICE with Put the 4 blocks inside your hands : 22) Red (taking 1)

 RANDOM CHOICE with Put the 4 blocks inside your hands : 22) Red (taking 1) Ans : 1 out of 4

 RANDOM CHOICE with Put the 4 blocks inside your hands : 23) Red or Blue(taking 1)

 RANDOM CHOICE with Put the 4 blocks inside your hands : 23) Red or Blue(taking 1) Ans : 2 out of 4 = 1 out of 2

 RANDOM CHOICE with Put the 4 blocks inside your hands : 24) Red then Blue(taking 2 without replacement)

 RANDOM CHOICE with Put the 4 blocks inside your hands : 24) Red then Blue(taking 2 without replacement) Ans : = 1 out of 12 RANDOM CHOICE with Put the 4 blocks inside your hands : 24) Red then Blue (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 25) Red (taking 1)

 RANDOM CHOICE with Put 8 blocks inside your hands : 25) Red (taking 1) Ans : 2 out of 8 = 1 out of 4

 RANDOM CHOICE with Put 8 blocks inside your hands : 26) Blue or Green (taking 1)

 RANDOM CHOICE with Put 8 blocks inside your hands : 26) Blue or Green (taking 1) Ans : 4 out of 8 = 1 out of 2

 RANDOM CHOICE with Put 8 blocks inside your hands : 27) Blue, Blue (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 27) Blue, Blue (taking 2 without replacement) Ans : 1 out of 28 RANDOM CHOICE with Put 8 blocks inside your hands : 27) Blue , Blue (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 28) Any 2 of the same colour (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 28) Any 2 of the same colour (taking 2 without replacement) Ans : 1 out of 7 RANDOM CHOICE with Put 8 blocks inside your hands : 28) Any 2 of the same colour (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 29) Any 2 of different colour (taking 2 without replacement)

 RANDOM CHOICE with Put 8 blocks inside your hands : 29) Any 2 of different colour (taking 2 without replacement) Ans : 6 out of 7 RANDOM CHOICE with Put 8 blocks inside your hands : 29) Any 2 of different colour (taking 2 without replacement)

 PERMUTATIONS with 30) Permutation question : How many ways can you order 8 blocks if there are 2 red, 2 blue, 2 yellow, 2 green ?

 PERMUTATIONS with 30) Permutation question : How many ways can you order 8 blocks if there are 2 red, 2 blue, 2 yellow, 2 green ? Ans : 8! / (2! 2! 2! 2!) = 2520 ways

 PERMUTATIONS with 31) Permutation question : How many ways can you order 8 blocks in a circle if there are 2 red, 2 blue, 2 yellow, 2 green ?

 PERMUTATIONS with 31) Permutation question : How many ways can you order 8 blocks in a circle if there are 2 red, 2 blue, 2 yellow, 2 green ? Ans : 7! / (2! 2! 2! 2!) = 315 ways

PERMUTATIONS with 32) Permutation question : How many ways can you select 2 blocks (order is important) from 5 blocks (green, green, blue, red, yellow) [note : 5P2 over 2 ! is not correct]

PERMUTATIONS with 32) Permutation question : How many ways can you select 2 blocks (order is important) from 5 blocks (green, green, blue, red, yellow) [note : 5P2 over 2 ! is not correct] Ans : 4P2 + 1 = 13 (see next slide)

PERMUTATIONS with 32) There is no difference between Green 1 and Green 2 So the permutations are : GB, GR, GY, BR, BG, BY, RG, RY, RB, YG, YB, YR. i.e. 4P2 = 12 + GG 1 Ans : 13

COMBINATIONS 33) Some groups have 4 blocks, and some groups have 4 triangular prisms. In how many ways can I choose one block and one triangular prism ?

COMBINATIONS with 33) Some groups have 4 blocks, and some groups have 4 triangular prisms. In how many ways can I choose one block and one triangular prism ? Ans : 16 ways (see next slide)

COMBINATIONS 33) Based on the Fundamental Counting Principle If I have m kind of things and n kind of things, I can combine one of each in m x n ways4 blocks x 4 triangular prisms = 16 ways Ans : 16 ways

LINKS Maths with Geoboards Hands on Lower Secondary Maths My Youtube channel : “ Maths with Geoboards “ ( for videos )My website : www.mathswithgeoboards.com ( for worksheets )My email : mathswithgeoboards@gmail.com ( for contact ) =========================================================================================================

Pythagoras’ Theorem Pythagoras Theorem Vol. 1 Pythagoras Theorem Vol. 2Proofs of Pythagoras Theorem - Geometric Proofs of Pythagoras Theorem - Algebraic Number Theory Prime Numbers Fibonacci Numbers Factors and Multiples GeneralHow to Make Geo-boards Angles Introductory Terms Easy Angle Problems The World’s Two Hardest Easy Geometry Problems IndicesIndices : Laws 1, 2, 3 and 4 Indices : Word ProblemsProbabilityThe Monty Hall Problem Use GeoboardsWorksheet Videos Youtube Channel : “ Maths with Geoboards” videos so far Worksheet VideosWorksheet Videos Use Geoboards

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PROBABILITY WORKSHOP with blocks By Jon - PowerPoint Presentation

PROBABILITY WORKSHOP with blocks By Jon - PPT Presentation

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