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## Chapter 3

### Part 1: For the problem in the Teacher's Edition, page 56

Make a list of all of the different ways that your class came up with to answer this question. You can provide them with the data sheet found on the Activity page (PDF file).

Sample Answer:
4 groups of 3 dancers and 5 groups of 2 dancers

#### Information

In the dance, there will always be a group of 4 or more dancers on stage and all of the dancers will be in rows. All of the rows must have at least 2 dancers in them and there must be the same number of dancers in each row. The choreographer wants to know how many different ways one group of dancers could be on stage and satisfy these requirements.

### Part 2: Be an Investigator

A good time to do this investigation is after Lesson 2 on prime and composite numbers.

#### Introducing the Investigation

Introduce the investigation by reading aloud the assignment on the first page of the Description of Investigation and Student Report (PDF file), by having one of your students read aloud the assignment, or by having the students read the assignment individually.

Make sure the students understand that a choreographer is someone who composed dances.

Put students in groups of two to four students to work on the investigation. Provide students with the Description of Investigation and Student Report (PDF file).

#### Doing the Investigation

The students may need help coming up with a strategy for solving this problem. The following plan shows one way to solve it.

• List all of the numbers from 4 through 12 that have factors other than the number itself and one. That list gives you all of the different numbers of dancers you can use in a group.
• Take each number (of dancers) and find all of its factors (other than the number itself and 1). Find all of the different ways you can multiply two numbers on the list of factors to get that number of dancers.

For example, 8 is a number that is on the list from 4 through 12. The factors of 8 other than 8 and 1 are 2 and 4. Both multiplying 2 by 4 and 4 by 2 gives you 8 dancers.

2 by 4 means 2 rows of 4 dancers.
4 by 2 means 4 rows of 2 dancers.

It might help students to draw pictures of the arrangement of dancers as they go along.

Answers:

group of 4 dancers:
2 by 2 arrangement
group of 6 dancers:
3 by 2 arrangement
2 by 3 arrangement
group of 8 dancers:
4 by 2
2 by 4
group of 9 dancers:
3 by 3
group of 10 dancers:
5 by 2
2 by 5
group of 12 dancers:
3 by 4
4 by 3
6 by 2
2 by 6

#### Student Report

The student report gives students an opportunity to write about what they have done, thus communicating about mathematics.

#### Extending the Investigation

Have students solve the same problem with a different number of dancers in the company, for example, 20.