Multiplication counting principle smartboard.notebook
August 18, 2010
Warm up:
1) 1/3 + 2/7
2) ­3/5 ­ 2/3
3) 5/9 4/7
4) 2/3 ÷ 3/8
Aug 18­9:35 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Aug 18­9:42 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Question:
What is the multiplication principle of counting? To what type of situation is it applied? Sep 25­10:38 AM
Launch:
1) Look at the menu above. List out all possible orders you could make consisting of exactly one entree and one side. Do not add a dessert. Sep 25­10:36 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
1) You receive some good news. You can add a dessert to your order. List all possible orders you can make with one entree and one side and one dessert? Again, a tree diagram is useful. You can add desserts to the diagram you used in part 1. C
BP
R
SV
BP
R
T
F
B
SV
BP
R
SV
BP
R SV
Sep 25­10:37 AM
3) In both questions above, the conjunction “and” was used. For example, you had a choice of one entree and one side and one dessert. How would the use of the conjunction “or” have changed the number of possible menu orders in question 2 above?
Sep 25­10:40 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Investigation:
Example 1: Choosing a vehicle
Suppose you are ready to buy your first automobile. You can only buy one vehicle. The first step in the task of selecting your vehicle is to determine the model. As a class or small group, narrow your choices down to 4 models. List them in the table below. As the second step of your vehicle selection process, you need to choose the exterior color. List 5 colors, you might choose for the exterior of your vehicle in the appropriate column of the table below. Sep 25­10:49 AM
How many ways are there for you to complete the first step (choice of model)?
How many ways are there for you to complete the second step (external color)? List all possible vehicles you could purchase by listing all possible model/external color combinations. You can choose to draw a tree diagram. How many different vehicle choices are there? L B m C
Sep 25­10:50 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
In the above setting, you were asked to choose both a model and an exterior color. What relationship do you notice between the number of total outcomes you counted and the number of choices you had for the two steps involved? Would the number of possibilities have been different if you had first chosen the color and then chosen the model? Explain. Sep 25­10:51 AM
Based on what you noticed above, if there had been 7 model choices and 4 color choices, how many model/color combinations would you have had available to you?
Write an expression for the total number of model and color combinations you would have had if there had been m vehicle models and n exterior colors to choose from. Here m and n can represent any whole number.
Sep 25­10:51 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
The relationship you have just discovered is known as the Multiplication Principle of Counting. This principle says that if a task involves two steps and the first step can be completed in m ways and the second step in n ways, then there are m*n ways to complete the task. This principle only applies in settings where the steps of the task are independent. For example, if some model choices had not had all color choices available then it would not have been possible to use this principle to count the vehicle choices. If the order of the steps is reversed in a two­step task where the steps are independent, does it change the number of ways to complete the task? What property of multiplication supports your answer? Sep 25­10:52 AM
Suppose you also have the chance to choose an interior color for our vehicle from Part 1. Recall you 4 models and 5 exterior colors to choose from. Additionally, you have 2 interior color choices. Add two interior color choices to the table below. Sep 25­10:52 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Again, consider the vehicle choice process as a two­step task such as is described in the Multiplication Principle of Counting. The first step is to choose model and exterior color. How many ways did you determine this could be done?
The second step is to choose interior color. How many ways are there to complete this second step?
According to the Multiplication Principle of Counting, how many ways are there to accomplish the task of choosing a model and exterior color followed by the task of choosing an interior color? Sep 25­10:52 AM
You just found a way to consider a three­step task as a two­
step task. In this way, you could apply the Multiplication Principle of Counting as it was presented. How did the total number of vehicle choices (model, exterior and interior color) relate to the number of choices for each of the threes steps in the task? Explain.
Write a version of the Multiplication Principle of Counting for a 3 step task by filling in the blank in the following statement. If a task involves three independent steps and the first step can be completed in m ways, the second step in n ways and the third step in p ways, then there are ________ ways to complete the task.
Sep 25­10:55 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Example 2: Pick a Number
You are given the instructions to write down a 3 digit number. You can only use the digits 0 through 9. You are allowed to use a digit more than once. (Thus a number like 882 is ok). The task of selecting your 3 digit number can be broken down into three steps, namely the selection of each digit. Fill in the following table:
Use the Multiplication Principle of Counting to determine the number of 3 digit numbers that can be made in this way.
Sep 25­10:56 AM
A restriction is added. You may not reuse a digit once it has been used. A number like 623 is still ok, but a number like 525 is not because the digit 5 is used twice. Should there be fewer or more possibilities now than in the previous problem? Explain.
Use the Multiplication Principle of Counting to determine the number of 3 digit numbers that can be made with the new restriction on reusing digits in place. Sep 25­10:57 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Example 3: Building a Sundae
Students are allowed to build an ice cream sundae. Each student starts with a bowl of vanilla ice cream. Then each goes through a sundae bar where there is a choice of any of the following toppings. The students can build their sundaes in any way they choose. This includes the option of no toppings at all. Below is the beginning of a list of toppings on this bar. Fill in the 3 blanks with toppings you would include on such a bar.
hot fudge, caramel, ­­_______________, ______________, and ____________. Sep 25­10:58 AM
For each student, the task of creating a sundae is a task involving five separate decisions. Below is the beginning of a tree diagram for building a sundae. The first two steps are filled in. Fill in the next branch of the diagram which is associated with step 3.
Step 1
Step 2
Step 3
Step 4
Sep 25­10:59 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
What sundae would you make? Can you see where that particular sundae choice would be represented on this tree diagram if it were completed?
The tree diagram shows how the process of building a sundae can be thought of as a multi­step task. How many steps are there in the task for this problem setting? What are the steps and how many choices do you have for each?
Sep 25­11:01 AM
Use the multiplication principle of counting to determine the number of possible sundaes the students can make at this sundae bar. Think of another topping you would like to see on the list. What is it? How many sundaes could be made with this additional topping on the bar? What is the obvious advantage in applying the multiplication principle of counting in a setting where there are multiple steps and many choices for each? Sep 25­11:01 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
Conclusions:
Based upon what you have learned in this investigation, you should now be able to answer the following questions.
In what type of situation is the multiplication principle of counting applied?
What does the multiplication principle of counting say about the number of ways to accomplish a two step task? In general, how can you apply the principle to a multi­step task?
Sep 25­11:02 AM
Sep 25­11:04 AM
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Multiplication counting principle smartboard.notebook
August 18, 2010
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