ITS PIZZA PARTY TIME

1
ITS PIZZA PARTY TIME ! ! ! WE GET TO PICK THE 2
TOPPINGS OF OUR CHOICE FROM A LIST OF TWO
DIFFERENT TOPPINGS. THE TOPPINGS WE GET TO
CHOOSE FROM ARE
PEPPERONI
ONIONS
USE YOUR RED AND YELLOW PATTERN SQUARES TO BUILD
YOUR PIZZAS.
2
ITS PIZZA PARTY TIME ! ! ! SO HOW MANY
DIFFERENT COMBINATIONS OF 2-TOPPING PIZZAS CAN WE
MAKE IF WE HAVE ONLY TWO DIFFERENT TOPPINGS TO
CHOOSE FROM?
PEPPERONI WITH ONIONS
ONIONS WITH PEPPERONI
OR, IF YOU PREFER
(THIS ONES EASY, RIGHT?)
3
SINCE PEPPERONI WITH ONIONS IS THE SAME EXACT
PIZZA AS ONIONS WITH PEPPERONI WE CANT COUNT
THAT COMBINATION TWICE SO WE CROSS IT OUT.
4
LETS DRAW A TREE DIAGRAM OF THIS . . .
1
WE CAN SEE THAT ONLY ONE 2-TOPPING PIZZA CAN BE
MADE.
5
ITS PIZZA PARTY TIME ! ! ! JUST HAVING ONE
PIZZA TO PICK FROM IS PRETTY BORINGSO LETS
EXPAND NOW WE GET TO PICK THE 2 TOPPINGS OF OUR
CHOICE FROM A LIST OF THREE DIFFERENT
TOPPINGS. THE THREE TOPPINGS WE GET TO CHOOSE
FROM ARE
PEPPERONI
ONIONS
MUSHROOMS
6
ITS PIZZA PARTY TIME ! ! ! HOW MANY DIFFERENT
COMBINATIONS OF 2-TOPPING PIZZAS CAN WE MAKE IF
WE HAVE THREE DIFFERENT TOPPINGS TO CHOOSE FROM?
LETS START BY CHOOSING PEPPERONI AS OUR FIRST
TOPPING THEN COMBINING PEPPERONI WITH EACH OF
THE OTHER TWO TOPPINGS. USE YOUR RED, YELLOW,
AND GREEN PATTERN SQUARES TO BUILD YOUR PIZZAS.
PEPPERONI WITH ONION
PEPPERONI WITH MUSHROOM
OR
7
LETS DRAW A TREE DIAGRAM OF WHAT WE HAVE SO FAR
. . .
1
2
USING PEPPERONI AS OUR FIRST TOPPING, WE FIND
THAT WE CAN COMBINE PEPPERONI WITH TWO OTHER
TOPPINGS FOR A TOTAL OF TWO POSSIBLE 2-TOPPING
PIZZAS. BUT WERE NOT DONE BECAUSE WE HAVE TWO
OTHER TOPPINGS TO CHOOSE FROM TO BE OUR FIRST
TOPPING.
8
ITS PIZZA PARTY TIME ! ! !
WHAT IF WE CHOOSE A DIFFERENT TOPPING OTHER THAN
PEPPERONI TO BE OUR FIRST TOPPING? LETS SEE
WHAT HAPPENS WHEN WE CHOOSE ONIONS TO BE OUR
FIRST TOPPING . . .
ONIONS WITH PEPPERONI
ONIONS WITH MUSHROOMS
OR
9
BUTSINCE PEPPERONI WITH ONIONS IS THE SAME
EXACT PIZZA AS ONIONS WITH PEPPERONI, WE
CANT COUNT THAT COMBINATION TWICE SO WE CROSS
IT OUT.
10
LETS DRAW A TREE DIAGRAM OF WHERE WE ARE NOW . .
.
1
2
3
WERE NOW UP TO THREE POSSIBLE 2-TOPPING PIZZAS.
BUT WERE NOT DONE YET BECAUSE WE HAVENT YET
CHOSEN OUR THIRD TOPPING, MUSHROOMS, TO BE OUR
FIRST TOPPING CHOICE.
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ITS PIZZA PARTY TIME ! ! !
LETS SEE WHAT HAPPENS WHEN WE CHOOSE OUR THIRD
AND FINAL TOPPING, MUSHROOMS, TO BE OUR FIRST
TOPPING . . .
MUSHROOMS WITH PEPPERONI
MUSHROOMS WITH ONION
OR
12
BUTSINCE MUSHROOMS WITH ONIONS IS THE SAME
EXACT PIZZA AS ONIONS WITH MUSHROOMS WE
CANT COUNT THAT COMBINATION TWICE SO WE CROSS
IT OUT.
13
AND ALSOSINCE MUSHROOMS WITH PEPPERONI IS
THE SAME EXACT PIZZA AS PEPPERONI WITH
MUSHROOMS WE CANT COUNT THAT COMBINATION
TWICE SO WE CROSS IT OUT.
14
LETS DRAW A TREE DIAGRAM OF WHERE WE ARE NOW . .
.
1
2
3
WE HAVE NOW USED EACH OF THE THREE TOPPINGS AS
OUR FIRST TOPPING CHOICE AND COMBINED EACH OF
THEM WITH EACH OF THE OTHER TOPPINGS AS OUR
SECOND CHOICE, MAKING SURE NOT TO COUNT THE SAME
COMBINATION TWICE.
WE CAN SEE THAT A TOTAL OF THREE DIFFERENT
2-TOPPING PIZZAS CAN BE MADE FROM A LIST OF
THREE TOPPING CHOICES.
15
STUDY THE TREE DIAGRAMS WE MADE . . .
THIS IS THE TREE DIAGRAM WE MADE WHEN WE BUILT
2-TOPPING PIZZAS CHOOSING FROM A LIST OF TWO
TOPPINGS.
1
THIS IS THE TREE DIAGRAM WE MADE WHEN WE BUILT
2-TOPPING PIZZAS CHOOSING FROM A LIST OF THREE
TOPPINGS.
1
2
3
16
DO YOU SEE A PATTERN?
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IF YOU SEE THE PATTERN . . . THEN YOU CAN
PROBABLY FIGURE OUT HOW MANY POSSIBLE 2-TOPPING
PIZZA COMBINATIONS ARE POSSIBLE WHEN YOU ADD A
FOURTH TOPPING TO YOUR LIST OF TOPPING CHOICES
WITHOUT HAVING TO DRAW A TREE DIAGRAM.
18
IF YOU DONT SEE THE PATTERN . . . THEN YOU CAN
STILL FIGURE OUT HOW MANY POSSIBLE 2-TOPPING
PIZZA COMBINATIONS ARE POSSIBLE WHEN YOU ADD A
FOURTH TOPPING TO YOUR LIST OF TOPPING CHOICES BY
BUILDING A TREE DIAGRAM.
19
FINDING PATTERNS IN COMBINATION PROBLEMS IS
POSSIBLE ONLY BY BEGINNING WITH TREE
DIAGRAMS. THEREFORE . . . SINCE MAKING TREE
DIAGRAMS IS SO IMPORTANT TO SOLVING COMBINATION
PROBLEMS YOUR ANSWER MUST CONSIST NOT ONLY OF A
NUMBER, BUT A TREE DIAGRAM AS WELL.
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WHEN YOU BUILD YOUR TREE DIAGRAM, REMEMBER THIS
. . .
CHOOSE PEPPERONI TO BE YOUR FIRST CHOICE, THEN
COMBINE IT WITH ALL THE OTHER CHOICES TO BE YOUR
SECOND CHOICE. NEXT, CHOOSE ONIONS TO BE YOUR
FIRST CHOICE, THEN COMBINE IT WITH ALL THE OTHER
CHOICES TO BE YOUR SECOND CHOICE. AND SO ON
BE CAREFUL TO CROSS OUT ANY DUPLICATE
COMBINATIONS AS YOU GO SO YOU DONT COUNT ONE
TWICE.
21
IF YOU DONT SEE THE PATTERN AND THE TREE
DIAGRAM IS CONFUSING TO YOU, TRY USING YOUR
PATTERN SQUARES TO ACTUALLY BUILD YOUR PIZZAS.
THEN, PERHAPS YOU CAN BUILD YOUR TREE DIAGRAM
FROM YOUR PIZZAS.
22
ITS PIZZA PARTY TIME ! ! ! CHOOSING 2 TOPPINGS
FROM A LIST OF FOUR DIFFERENT TOPPINGS
PEPPERONI
ONIONS
MUSHROOMS
BLACK OLIVES
HOW MANY 2-TOPPING PIZZA COMBINATIONS ARE
POSSIBLE IF THERE ARE FOUR TOPPINGS TO CHOOSE
FROM?
23
REMEMBER . . . YOUR ANSWER MUST CONSIST NOT ONLY
OF A NUMBER, BUT A TREE DIAGRAM AS WELL.
24
ITS PIZZA PARTY TIME ! ! ! CHOOSING 2 TOPPINGS
FROM A LIST OF FOUR DIFFERENT TOPPINGS
PEPPERONI
ONIONS
MUSHROOMS
BLACK OLIVES
HOW MANY 2-TOPPING PIZZA COMBINATIONS ARE
POSSIBLE IF THERE ARE FOUR TOPPINGS TO CHOOSE
FROM?