Deciding Whether Order Is Important

Clever Counting Investigation 4

The bigger problem…

Detective Curious considered the possibility that the locker robbery may have taken place while the security guard and her friend were playing dominoes. She reasoned that they may have been so involved in their game that they would not have noticed a disturbance among the lockers.

• Have you ever played a game of dominoes?

• How else have you played with dominoes?

• What’s another name you’ve heard for dominoes? Why do you think that is?

Pips

The dots on a domino are called pips.

Double-Two Set

• A set of dominoes in which the highest pip value for half of the domino is 2, is called a double-two set.

• Which ones fit this set?

• How many dominoes are in a double-two set?

Double-Six Set

• The security guard was playing with a standard set of dominoes, which is a double-six set.

• How many dominoes do you think make up a double-six set?

??????

Problem 4.1

A. How many different dominoes are in a complete set? How do you know for sure?

B. The vending machines at Fail-Safe offer seven types of sandwiches and seven different drinks. The security guard wants to buy one sandwich and one drink. From how many combinations can she choose?

C. The security guard in a nearby storage warehouse can follow seven routes from checkpoint A to checkpoint B and seven routes from checkpoint B to checkpoint C. How many routes can he follow from checkpoint A to C through checkpoint B?

D. Parts A-C each involve finding the number of ways to fill two positions when there are seven choices for each position. Compare the strategies you used to answer each part. How are the strategies similar? How are they different?

Teacher Hat

• Goals– To identify the difference in the structure of

problems in which order is not important from those in which it is

– To create a model to clarify a situation– To generalize a pattern

Choosing LocksProblem 4.2

• Begins by discussing situations when order does and does not matter– Dominoes

– Zip Codes

– “Combination” Locks vs. “Permutation” Locks

• Parts A and B ask students to choose two (A) or three (B) different locks from six choices. – List all combinations without repetition

• Realizing that ABC is the same as BAC, CAB…

As a group, design a lesson for problem 4.2.

1. Work through the problem first as students.

2. Design a launch, explore, summarize.

3. Assign homework from the ACE questions.