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4-7, 13+

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## Using Concrete Manipulatives to Generate Algebraic Patterns

Using manipulatives, students figure out patterns in problems involving seating arrangements. They write algebraic equations to describe the relationships.

### Overview

In this Cyberchase activity, students are given three variations of a problem involving a seating arrangement using rectangular tables. Using manipulatives, they must figure out the patterns relating the number of tables and seats. In each case, they write an algebraic equation that describes the relationship.

4-7

### Suggested Time

60 minutes

### Media Resources

Arrangement 1 QuickTime Video
Arrangement 2 QuickTime Video
Arrangement 3 QuickTime Video

### Part I: Learning Activity

1. Read the following to your students: In this Cyberchase video segment, Archimedes is going to be the grandmaster for the parade which celebrates Starlight Night. To run the generator that lights the Cyberstars, 20 circuit boards are required. Twenty workers together must complete each circuit board. Hacker kidnaps Archimedes, so the CyberSquad must figure out how to organize the 20 workers to complete the assembly. They try three arrangements to solve the problem.

2. Distribute the Sitting Around the Table handout.

3. Ask the students to work on Part I of the handout.

4. Tell the students that they will now watch the first video segment in which the CyberSquad tries to work out one arrangement of the table and chairs.

5. Show the students Arrangement 1 QuickTime Video . Ask students to focus on how the number of seats for workers changes as the number of tables grows.

6. Briefly discuss the students' work on Part I of the handout.

7. Read the following to the students: "When the first arrangement doesn't allow the workers to build each circuit board in an orderly sequence, the CyberSquad decides to arrange the tables side-by-side. In this arrangement, each table is placed right next to the other tables. For each table side that is exposed, one worker sits at the table. Fill in the values in the column on the right until you have enough places for all 20 workers. Look at the picture provided. It illustrates this seating arrangement for one and two tables.

8. Ask the students to work Part II on the handout.

9. Tell the students that they will now watch the next video segment in which the CyberSquad works on the second table and chair arrangement.

10. Show the students Arrangement 2 QuickTime Video .

11. Briefly discuss the students' work on Part II of the handout; especially try to elicit different ways of explaining why the second arrangement did not work for all twenty workers.

12. Read the following to the students: "Because there aren't enough tables to seat all 20 workers in the previous arrangement, the Cyberchase kids try one more arrangement with the tables end-to-end. In this arrangement, each table is placed next to another table with the short sides placed together. For each long side that is exposed, two workers sit at the table. No workers sit on the short sides. Fill in the following table of values until you have enough places for all 20 workers. Look at the picture which illustrates this seating arrangement for one and two tables."

13. Ask the students to complete Part III on the handout.

14. Tell the students that they will now watch the final video segmentof this activity, in which the CyberSquad attempts to solve the final table arrangement.

15. Play the Arrangement 3 QuickTime Video .

16. Discuss the differences among the three arrangements. Discuss why the final arrangement provides a solution to the problem the CyberSquad faces.

### Part II: Assessment

Assessment: Level A (proficiency): Students are asked to write an algebraic equation for another seating arrangement.

Assessment: Level B (above proficiency): Students extend their knowledge of algebraic equations to deal with a pattern of squares that increase.

## Educational Standards

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