8. Clever Counting - CMS Goals

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Recognize situations in which counting techniques apply

Construct organized lists of outcomes for complex processes and uncover patterns that help in counting the outcomes of those processes

Use diagrams, tables, and symbolic expressions to organize examples in listing and counting tasks

Analyze the usefulness of counting trees and use counting trees

Use mental arithmetic to make estimates in multiplication and division calculations

Analyze counting problems involving choices in various contexts

Differentiate among situations in which order does and does not matter and in which repeats ate and are not allowed

Analyze the number of paths through a network

Compare the structures of networks with problems involving combinations

Create networks that satisfy given constraints

Apply thinking and reasoning skills to an open-ended situation in which assumptions must be made and create a persuasive argument to support a conjecture


1. Counting Possibilities 

     1.1     Making Faces

     1.2    Checking Plate Numbers

               ACE Questions  (1 - 19 )

              

To analyze counting problems involving choices in different contexts.

To analyze the usefulness of counting trees

To use counting trees

To begin to see a connections between some counting situations and the operation of multiplication.

2.    Opening Locks

     2.1    Pushing Buttons

     2.2    Dialing Combinations

     2.3    Increasing Security

               ACE Questions ( 1 - 14 )

To further explore counting situations in which multiplication provides an answer

To construct systematic lists of outcomes for complex processes

To uncover patterns that help in counting the outcomes of complex processes

To recognizes that one problem has the same structure as another problem

3.    Networks 

     3.1    Making Rounds

     3.2    Networking

     3.3    Designing Networks

               ACE Questions  ( 1 - 10 )

To explore network applications

To analyze the number of paths in a network

To compare the structures of networks and problems involving combinations

To create networks that satisfy given constraints

4.    Deciding w Order is Important

     4.1    Playing Dominoes

      4.2    Choosing Locks(A 1 - 14 )

To identify the difference in the structure of problems in which order is not important form those in which it is

To create a model to clarify a situation

To generalize a pattern

5.    Wrapping Things Up

     5.1    Catching a Bicycle Thief

               ACE Questions  ( 1 - 17 )

To recognize situations in which counting techniques apply

To differentiate among situations in which order does not matter and in which repeats are and are not allowed

To use a variety of models to clarify a solutions

To consider situations in which counting techniques would not apply

To apply thinking and reasoning skills to an open-ended situation in which assumptions must be made and to create a persuasive argument to support a conjecture